## 13.1: Iterated Integrals and Area

### Terms and Concepts

1. When integrating (f_x(x,y)) with respect to *x*, the constant of integration *C* is really which: (C(x) ext{ or }C(y))? What does this mean?

2. Integrating an integral is called _________ __________.

3. When evaluating an iterated integral, we integrate from _______ to ________, then from _________ to __________.

4. One understanding of an iterated integral is that (displaystyle int_a^b int_{g_1(x)}^{g_2(x)},dy,dx) gives the _______ of a plane region.

### Problems

**In Exercises 5-10, evaluate the integral and subsequent iterated integral.**

5.

(a) (displaystyle int_2^5 (6x^2+4xy-3y^2),dy)

(b) (displaystyle int_{-3}^2 int_2^5 (6x^2+4xy-3y^2),dy,dx)

6.

(a) (displaystyle int_0^pi (2xcos y +sin x),dx)

(b) (displaystyle int_{0}^{pi/2} int_0^pi (2xcos y +sin x),dx,dy)

7.

(a) (displaystyle int_1^x (x^2y-y+2),dy)

(b) (displaystyle int_0^2 int_1^x (x^2y-y+2),dy,dx)

8.

(a) (displaystyle int_y^{y^2} (x-y),dx)

(b) (displaystyle int_{-1}^1 int_y^{y^2} (x-y),dx,dy)

9.

(a) (displaystyle int_0^{y} (cos x sin y),dx)

(b) (displaystyle int_0^pi int_0^{y} (cos x sin y),dx,dy)

10.

(a) (displaystyle int_0^{x} left (frac{1}{1+x^2}
ight ),dy)

(b) (displaystyle int_1^2 int_0^{x} left (frac{1}{1+x^2}
ight ),dy,dx)

**In Exercises 11-16, a graph of a planar region (R) is given. Give the iterated integrals, with both orders of integration (dy,dx) and (dx,dy), that give the area of (R). Evaluate one of the iterated integrals to find the area.**

11.

12.

13.

14.

15.

16.

**In Exercises 17-22, iterated integrals are given that compute the area of a region R in the (xy)-plane. Sketch the region R, and give the iterated integral(s) that give the area of R with the opposite order of integration.**

17. (displaystyle int_{-2}^2 int_0^{4-x^2},dy,dx)

18. (displaystyle int_{0}^1 int_{5-5x}^{5-5x^2},dy,dx)

19. (displaystyle int_{-2}^2 int_0^{2sqrt{4-y^2}},dx,dy)

20. (displaystyle int_{-3}^3 int_{-sqrt{9-x^2}}^{sqrt{9-x^2}},dy,dx)

21. (displaystyle int_{0}^1 int_{-sqrt{y}}^{sqrt{y}},dx,dy +int_1^4 int_{y-2}^{sqrt{y}},dx,dy)

22. (displaystyle int_{-1}^1 int_{(x-1)/2}^{(1-x)/2},dy,dx)

## 13.2: Double Integration and Volume

### Terms and Concepts

1. An integral can be interpreted as giving the signed area over an interval; a double integral can be interpreted as giving the signed ________ over a region.

2. Explain why the following statement is false: "Fubini's Theorem states that (int_a^b int_{g_1(x)}^{g_2(x)}f(x,y),dy,dx = int_a^b int_{g_1(y)}^{g_2(y)}f(x,y),dx,dy)."

3. Explain why if (f(x,y)>0) over a region *R*, then (intint_R f(x,y),dA >0).

4. If (intint_R f(x,y)dA= intint_R g(x,y),dA), does this imply (f(x,y)=g(x,y))?

### Problems

**In Exercises 5-10,(a) Evaluate the given iterated integral, and(b) rewrite the integral using the other order of integration.**

5. (int_1^2 int_{-1}^1 left ( frac{x}{y}+3 ight ),dx,dy)

6. (int_{-pi/2}^{pi/2} int_{0}^pi (sin x cos y),dy,dx)

7. (int_0^4 int_{0}^{-x/2+2} left ( 3x^2-y+2 ight ),dy,dx)

8. (int_1^3 int_{y}^3 left ( x^2y-xy^2 ight ),dx,dy)

9. (int_0^21int_{-sqrt{1-y}}^{sqrt{1-y}}( x+y+2 ),dx,dy)

10. (int_0^9 int_{y/3}^{sqrt{3}} left ( xy^2 ight ),dx,dy)

**In Exercises 11-18:(a) Sketch the region R given by the problem.(b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region R.(c) Evaluate one of the iterated integrals to find the signed volume under the surface **(z=f(x,y))

**over the region**

*R.*11. (intint_R x^2y,dA), where *R* is bounded by (y=sqrt{x} ext{ and }y=x^2).

12. (intint_R x^2y,dA), where *R* is bounded by (y=sqrt[3]{x} ext{ and }y=x^3).

13. (intint_R x^2-y^2,dA), where *R* is the rectangle with corners ((-1,-1),(1,-1),(1,1) ext{ and }(-1,1)).

14. (intint_R ye^x,dA), where *R* is bounded by (x=0,,x=y^2 ext{ and }y=1).

15. (intint_R (6-3x-2y),dA), where *R* is bounded by (x=0,y=0 ext{ and }3x+2y=6).

16. (intint_R e^y,dA), where *R* is bounded by (y=ln x ext{ and }y=frac{1}{e-1}(x-1)).

17. (intint_R (x^3y-x),dA), where *R* is the half of the circle (x^2+y^2=9) in the first and second quadrants.

18. (intint_R (4-sy),dA), where *R* is bounded by (y=0,y=x/e ext{ and }y=ln x).

**In Exercises 19-22, state why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral.**

19. (int_0^4 int_{y/2}^2 e^{x^2},dx,dy)

20. (int_0^{sqrt{pi/2}} int_{x}^{sqrt{pi/2}} cos (y^2),dy,dx)

21. (int_0^1 int_{y}^1 frac{2y}{x^2+y^2},dx,dy)

22. (int_{-1}^1 int_{1}^2 frac{x an^2 y}{1+ln y},dy,dx)

**In Exercises 23-26, find the average value of f over the region R. Notice how these functions and regions are related to the iterated integrals given in Exercises 5-8.**

23. (f(x,y)=frac{x}{y}+3); *R* is the rectangle with opposite corners ((-1,1) ext{ and }(1,2)).

24. (f(x,y)=sin x cos y); *R* is bounded by (x=0,x=pi,y=-pi/2 ext{ and }y=pi/2).

25. (f(x,y)=3x^2-y+2); *R* is bounded by the lines (y=0,y=2-x/2 ext{ and }x=0).

26. (f(x,y)=x^2y-xy^2); *R* is bounded by (y=x,y=1 ext{ and }x=3).

## 13.3: Double Integration with Polar Coordinates

### Terms and Concepts

1. When evaluating (intint_R f(x,y),dA) using polar coordinates, (f(x,y)) is replaced with _______ and (dA) is replaced with _______.

2. Why would one be interested in evaluating a double integral with polar coordinates?

### Problems

**In Exercises 3-10, a function** (f(x,y))** is given and a region R of the x-y plane is described. Set up and evaluate** (intint_R f(x,y),dA)

**.**

3. (f(x,y)=3x-y+4); *R* is the region enclosed by the circle (x^2+y^2=1).

4. (f(x,y)=4x+4y); *R* is the region enclosed by the circle (x^2+y^2=4).

5. (f(x,y)=8-y); *R* is the region enclosed by the circles with polar equations (r=cos heta ext{ and }r=3cos heta).

6. (f(x,y)=4); *R* is the region enclosed by the petal of the rose curve (r=sin (2 heta)) in the first quadrant.

7. (f(x,y)=ln (x^2+y^2)); *R* is the annulus enclosed by the circles (x^2+y^2=1 ext{ and }x^2+y^2=4.

8. (f(x,y)=1-x^2-y^2); *R* is the region enclosed by the circle (x^2+y^2=1).

9. (f(x,y)=x^2-y^2); *R* is the region enclosed by the circle (x^2+y^2=36) in the first and fourth quadrants.

10. (f(x,y)=(x-y)/(x+y)); *R* is the region enclosed by the lines (y=x,y=0) and the circle (x^2+y^2=1) in the first quadrant.

**In Exercises 11-14, an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral.**

11. (int_0^5 int_{-sqrt{25-x^2}}^{sqrt{25-x^2}}sqrt{x^2+y^2}dy,dx)

12. (int_{-4}^4 int_{-sqrt{16-y^2}}^{0}(2y-x)dx,dy)

13. (int_0^2 int_{y}^{sqrt{8-y^2}}(x+y),dx,dy)

14. (int_{-2}^{-1} int_{0}^{sqrt{4-x^2}}(x+5)dy,dx+int_{-1}^1int_{sqrt{1-x^2}}^{sqrt{4-x^2}}(x+5),dy,dx+int_1^2int_0^{sqrt{4-x^2}}(x+5),dy,dx)

**In Exercises 15-16, special double integrals are presented that are especially well suited for evaluation in polar coordinates.**

15. Consider (intint_R e^{-(x^2+y^2)}dA.)

(a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region *R*?

(b) Let *R* be the region bounded by the circle of radius *a* centered at the origin. Evaluate the double integral using polar coordinates.

(c) Take the limit of your answer from (b), as (a o infty). What does this imply about the volume under the surface of (e^{-(x^2+y^2)}) over the entire *x-y* plane?

16. The surface of a right circular cone with height *h* and base radius *a* can be described by the equation (f(x,y)=h-hsqrt{frac{x^2}{a^2}+frac{y^2}{a^2}}), where the tip of the cone lies at ((0,0,h)) and the circular base lies in the *x-y* plane, centered at the origin.

Confirm that the volume of a right circular cone with height *h* and base radius *a* is (V=frac{1}{3}pi a^2h) by evaluating (intint_R f(x,y),dA) in polar coordinates.

## 13.4: Center of Mass

### Terms and Concepts

1. Why is it easy to use "mass" and "weight" interchangeably, even though they are different measures?

2. Given a point ((x,y)), the value of *x* is a measure of distance from the _________-axis.

3. We can think of (intint_R dm) as meaning "sum up lots of ________."

4. What is a "discrete planar system?"

5. Why does (M_x) use (intint_R ydelta (x,y),dA) instead of (intint_R xdelta (x,y),dA); that is, why do we use "y" and not "x"?

6. Describe a situation where the center of mass of a lamina does not lie within the region of the lamina itself.

### Problems

**In Exercises 7-10, point masses are given along a line or in the plane. Find the center of mass **(overline{x})** or **((overline{x},overline{y}))**, as appropriate. (All masses are in grams and distances are in cm.)**

7. (m_1 =4 ext{ at }x=1;quad m_2=3 ext{ at }x=3;quad m_3 = 5 ext{ at }x=10)

8. (m_1 =2 ext{ at }x=-3;quad m_2=2 ext{ at }x=-1;quad m_3 = 3 ext{ at }x=0;quad m_4=3 ext{ at }x=7)

9. (m_1 =2 ext{ at }(-2,2);quad m_2=2 ext{ at }(2,-2);quad m_3 = 20 ext{ at }(0,4))

10. (m_1 =1 ext{ at }(-1,1);quad m_2=2 ext{ at }(-1,1);quad m_3 = 2 ext{ at }(1,1);quad m_4 =1 ext{ at }(1,-1))

**In Exercises 11-18, find the mass/weight of the lamina described by the region R in the plane and its density function **(delta (x,y)).

11.* R* is the rectangle with corners ((1,-3),(1,2),(7,2) ext{ and }(7,-3);delta (x,y)=5)gm/cm(^2)

12.* R* is the rectangle with corners ((1,-3),(1,2),(7,2) ext{ and }(7,-3);delta (x,y)=(x+y^2))gm/cm(^2)

13.* R* is the triangle with corners ((-1,0),(1,0), ext{ and }(0,1);delta (x,y)=2)lb/in(^2)

14. *R* is the triangle with corners ((0,0),(1,0), ext{ and }(0,1);delta (x,y)=(x^2+y^2+1))lb/in(^2)

15. *R* is the circle centered at the origin with radius 2; (delta (x,y)=(x+y+4))kg/m(^2)

16. *R* is the circle sector bounded by (x^2+y^2=25) in the first quadrant; (delta (x,y) =(sqrt{x^2+y^2}+1))kg/m(^2)

17. *R* is the annulus in the first and second quadrants bounded by (x^2+y^2=9 ext{ and }x^2+y^2=36;delta (x,y)=4)lb/ft(^2)

18. *R* is the annulus in the first and second quadrants bounded by (x^2+y^2=9 ext{ and }x^+y^2=36;delta (x,y)=sqrt{x^2+y^2})lb/ft(^2)

**In Exercises 19-26, find the center of mass of the lamina described by the region R in the plane and its density function** (delta (x,y))

**.**

**Note: these are the same lamina as in Exercises 11-18.**

19.* R* is the rectangle with corners ((1,-3),(1,2),(7,2) ext{ and }(7,-3);delta (x,y)=5)gm/cm(^2)

20.* R* is the rectangle with corners ((1,-3),(1,2),(7,2) ext{ and }(7,-3);delta (x,y)=(x+y^2))gm/cm(^2)

21.* R* is the triangle with corners ((-1,0),(1,0), ext{ and }(0,1);delta (x,y)=2)lb/in(^2)

22. *R* is the triangle with corners ((0,0),(1,0), ext{ and }(0,1);delta (x,y)=(x^2+y^2+1))lb/in(^2)

23. *R* is the circle centered at the origin with radius 2; (delta (x,y)=(x+y+4))kg/m(^2)

24. *R* is the circle sector bounded by (x^2+y^2=25) in the first quadrant; (delta (x,y) =(sqrt{x^2+y^2}+1))kg/m(^2)

25. *R* is the annulus in the first and second quadrants bounded by (x^2+y^2=9 ext{ and }x^2+y^2=36;delta (x,y)=4)lb/ft(^2)

26. *R* is the annulus in the first and second quadrants bounded by (x^2+y^2=9 ext{ and }x^+y^2=36;delta (x,y)=sqrt{x^2+y^2})lb/ft(^2)

**The moment of inertia **(i)** is a measure of the tendency of lamina to resist rotating about an axis or continue to rotate about an axis. **(i_x)** is the moment of inertia about the x-axis, **(i_x)** is the moment of inertia about the x-axis, and **(i_o)** is the moment of inertia about the origin.These are computed as follows:**

- (i_x = intint_R y^2,dm)
- (i_y = intint_R x^2,dm)
- (i_o = intint_R (x^2+y^2),dm)

**In Exercises 27-30, a lamina corresponding to a planar region R is given with a mass of 16 units. For each, compute **(i_x), (i_y)

**and**(i_o)

**.**

27. *R *is the 4 x 4 square with corners ((-2,-2) ext{ and }(2,2)) with density (delta (x,y)=1).

28. *R *is the 8 x 2 rectangle with corners ((-4,-1) ext{ and }(4,1)) with density (delta (x,y)=1).

29. *R *is the 4 x 2 rectangle with corners ((-2,-1) ext{ and }(2,1)) with density (delta (x,y)=2).

30. *R *is the circle with radius 2 centered at the origin with density (delta (x,y)=4/pi).

## 13.5: Surface Area

### Terms and Concepts

1. "Surface area" is analogous to what previously studied concept?

2. To approximate the area of a small portion of a surface, we computed the area of its ______ plane.

3. We interpret (intint_R ,dS) as "sum up lots of little _______ ________."

4. Why is it important to know how to set up a double integral to compute surface area, even if the resulting integral is hard to evaluate?

5. Why do (z=f(x,y)) and (z=g(x,y)=f(x,y)+h), for some real number *h*, have the same surface area over a region *R*?

6. Let (z=f(x,y) ) and (z=g(x,y)=2f(x,y)). Why is the surface area of *g* over a region *R* not twice the surface area of (f) over (R)?

### Problems

**In Exercises 7-10, set up the iterated integral that computes the surfaces area of the given surface over the region R.**

7. (f(x,y)=sin x cos y;quad R) is the rectangle with bounds (0le xle 2pi), (0le y le 2pi).

8. (f(x,y)=frac{1}{x^2+y^2+1};quad R) is the circle (x^2+y^2=9).

9. (f(x,y)=x^2-y^2;quad R) is the rectangle with opposite corners ((-1,-1)) and (1,1)).

10. (f(x,y)=frac{1}{e^{x^2}+1};quad R) is the rectangle bounded by (-5le x le 5) and (0le y le 1).

**In Exercises 11-19, find the area of the given surface over the region R.**

11. (f(x,y)=3x-7y+2;quad R) is the rectangle with opposite corners ((-1,0) ext{ and }(1,3)).

12. (f(x,y)=2x+2y+2;quad R) is the triangle with corners ((0,0),(1,0) ext{ and }(0,1)).

13. (f(x,y)=x^2+y^2+10;quad R) is the circle (x^2+y^2=16).

14. (f(x,y)=-2x+4y^2+7 ext{ over } R), the triangle bounded by (y=-x,y=x,0le y le 1).

15. (f(x,y)=x^2+y) over *R*, the triangle bounded by (y=2x,y=0 ext{ and }x=2).

16. (f(x,y)=frac{2}{3}x^{3/2}) over *R*, the rectangle with opposite corners ((0,0) ext{ and }(1,1)).

17. (f(x,y)=10-2sqrt{x^2+y^2}) over *R*, the circle (x^2+y^2=25). (This is the cone with height 10 and base radius 5; be sure to compare your result with the known formula.)

18. Find the surface area of the sphere with radius 5 by doubling the surface area of (f(x,y)=sqrt{25-x^2-y^2}) over *R*, the circle (x^2+y^2=25). (Be sure to compare your result with the known formula.)

19. Find the surface area of the ellipse formed by restricting the plane (f(x,y)=cx+dy+h) to the region *R*, the circle (x^2+y^2=1), where *c, d* and *h* are some constants. Your answer should be given in terms of *c* and *d*; why does the value of *h* not matter?

## 13.6: Volume Between Surfaces and Triple Integration

### Terms and Concepts

1. The strategy for establishing bounds for triple integrals is "________ to ________, _________ and __________ to _______."

2. Give an informal interpretation of what ("intintint_D ,dV)" means.

3. Give two uses of triple integration.

4. If an object has a constant density (delta) and a volume *V*, what is its mass?

### Problems

**In Exercises 5-8, two surfaces **(f_1(x,y))** and **(f_2(x,y))** and a region R in the x,y plane are given. Set up and evaluate the double integral that finds the volume between these surfaces over R.**

5. (f_x(x,y) = 8-x^2-y^2,,f_2(x,y) =2x+y;)*R* is the square with corners ((-1,-1) ext{ and }(1,1)).

6. (f_x(x,y) = x^2+y^2,,f_2(x,y) =-x^2-y^2;)*R* is the square with corners ((0,0) ext{ and }(2,3)).

7. (f_x(x,y) = sin x cos y,,f_2(x,y) =cos x sin y +2;)*R* is the triangle with corners ((0,0), (pi , 0) ext{ and }(pi,pi)).

8. (f_x(x,y) = 2x^2+2y^2+3,,f_2(x,y) =6-x^2-y^2;)*R* is the circle (x^2+y^2=1).

**In Exercises 9-16, a domain D is described by its bounding surfaces, along with a graph. Set up the triple integrals that give the volume of D in all 6 orders of integration, and find the volume of D by evaluating the indicated triple integral.**

9. *D* is bounded by the coordinate planes and (z=2-2x/3-2y).

Evaluate the triple integral with order *dz dy dz*.

10. *D* is bounded by the planes (y=0,y=2,x=1,z=0 ext{ and }z=(2-x)/2).

Evaluate the triple integral with order *dx dy dz*.

11. *D* is bounded by the planes (x=0,x=2,z=-y ext{ and by }z=y^2/2).

Evaluate the triple integral with order *dy dz dx*.

12. *D* is bounded by the planes (z=0,y=9, x=0 ext{ and by )z=sqrt{y^2-9x^2}).

Do not evaluate any triple integral.

13. *D* is bounded by the planes (x=2,y=1,z=0 ext{ and }z=2x+4y-4).

Evaluate the triple integral with order *dx dy dz*.

14. *D* is bounded by the plane (z=2y ext{ and by }y=4-x^2).

Evaluate the triple integral with order *dz dy dz*.

15. *D* is bounded by the coordinate planes and (y=1-x^2 ext{ and }y=1-z^2).

Do not evaluate any triple integral. Which order is easier to evaluate: *dz dy dx *or *dy dz dx*? Explain why.

16. *D* is bounded by the coordinate planes and by (z=1-y/3 ext{ and }z=1-x).

Evaluate the triple integral with order *dx dy dz*.

**In Exercises 17-20, evaluate the triple integral.**

17. (int_{-pi/2}^{pi/2}int_{0}^{pi}int_{0}^{pi} (cos x sin y sin z )dz,dy,dx)

18. (int_{0}^{1}int_{0}^{x}int_{0}^{x+y} (x+y+z )dz,dy,dx)

19. (int_{0}^{pi}int_{0}^{1}int_{0}^{z} (sin (yz))dx,dy,dz)

20. (int_{pi}^{pi^2}int_{x}^{x^3}int_{-y^2}^{y^2} (cos x sin y sin z )dz,dy,dx)

**In Exercises 21-24, find the center of mass of the solid represented by the indicated space region D with density function** (delta (x,y,z)).

21. *D* is bounded by the coordinate planes and (z=2-2x/3-2y); (delta (x,y,z)=10)g/cm(^3).

(Note: this is the same region as used in Exercise 9.)

22. *D* is bounded by the planes (y=0,y=2,x=1,z=0 ext{ and }z=(3-x)/2); (delta (x,y,z)=2)g/cm(^3).

(Note: this is the same region as used in Exercise 10.)

23. *D* is bounded by the planes (x=2,y=1,z=0 ext{ and }z=2x+4y-4); (delta (x,y,z)=x^2)lb/in(^3).

(Note: this is the same region as used in Exercise 13.)

24. *D* is bounded by the planes (z=2y ext{ and by }y=4-x^2). (delta (x,y,z)=y^2)lb/in(^3).

(Note: this is the same region as used in Exercise 14.)

#### Chapter 14, Problem Review_Exercises_Section_i 9

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## Details

### Fabrizio Gabbiani

Dr. Gabbiani is Professor in the Department of Neuroscience at the Baylor College of Medicine. Having received the prestigious Alexander von Humboldt Foundation research prize in 2012, he just completed a one-year cross appointment at the Max Planck Institute of Neurobiology in Martinsried and has international experience in the computational neuroscience field. Together with Dr. Cox, Dr. Gabbiani co-authored the first edition of this bestselling book in 2010.

### Affiliations and Expertise

Baylor College of Medicine, Houston, TX, USA

### Steven Cox

Dr. Cox is Professor of Computational and Applied Mathematics at Rice University. Affiliated with the Center for Neuroscience, Cognitive Sciences Program, and the Ken Kennedy Institute for Information Technology, he is also Adjunct Professor of Neuroscience at the Baylor College of Medicine. In addition, Dr. Cox has served as Associate Editor for a number of mathematics journals, including the Mathematical Medicine and Biology and Inverse Problems. He previously authored the first edition of this title with Dr. Gabbiani.

### Affiliations and Expertise

Computational and Applied Mathematics, Rice University, Houston, TX, USA

## Table of Contents

**I. A LIBRARY OF ELEMENTARY FUNCTIONS**

**1. Linear Equations and Graphs**

1.1 Linear Equations and Inequalities

Chapter 1 Summary and Review

**2. Functions and Graphs**

2.2 Elementary Functions: Graphs and Transformations

2.4 Polynomial and Rational Functions

Chapter 2 Summary and Review

**II. FINITE MATHEMATICS**

**3. Mathematics of Finance**

3.2 Compound and Continuous Compound Interest

3.3 Future Value of an Annuity Sinking Funds

3.4 Present Value of an Annuity Amortization

Chapter 3 Summary and Review

**4. Systems of Linear Equations Matrices**

4.1 Review: Systems of Linear Equations in Two Variables

4.2 Systems of Linear Equations and Augmented Matrices

4.3 Gauss&ndashJordan Elimination

4.4 Matrices: Basic Operations

4.5 Inverse of a Square Matrix

4.6 Matrix Equations and Systems of Linear Equations

4.7 Leontief Input&ndashOutput Analysis

Chapter 4 Summary and Review

**5. Linear Inequalities and Linear Programming**

5.1 Linear Inequalities in Two Variables

5.2 Systems of Linear Inequalities in Two Variables

5.3 Linear Programming in Two Dimensions: A Geometric Approach

Chapter 5 Summary and Review

**6. Linear Programming: The Simplex Method**

6.1 The Table Method: An Introduction to the Simplex Method

6.2 The Simplex Method: Maximization with Problem Constraints of the Form ≤

6.3 The Dual Problem: Minimization with Problem Constraints of the Form ≥

6.4 Maximization and Minimization with Mixed Problem Constraints

Chapter 6 Summary and Review

**7. Logic, Sets, and Counting**

7.3 Basic Counting Principles

7.4 Permutations and Combinations

Chapter 7 Summary and Review

8.1 Sample Spaces, Events, and Probability

8.2 Union, Intersection, and Complement of Events Odds

8.3 Conditional Probability, Intersection, and Independence

8.5 Random Variable, Probability Distribution, and Expected Value

Chapter 8 Summary and Review

**9. Limits and the Derivative**

9.1 Introduction to Limits

9.2 Infinite Limits and Limits at Infinity

9.5 Basic Differentiation Properties

9.7 Marginal Analysis in Business and Economics

Chapter 9 Summary and Review

**10. Additional Derivative Topics**

10.1 The Constant *e* and Continuous Compound Interest

10.2 Derivatives of Exponential and Logarithmic Functions

10.3 Derivatives of Products and Quotients

10.5 Implicit Differentiation

Chapter 10 Summary and Review

**11. Graphing and Optimization**

11.1 First Derivative and Graphs

11.2 Second Derivative and Graphs

11.4 Curve-Sketching Techniques

11.5 Absolute Maxima and Minima

Chapter 11 Summary and Review

12.1 Antiderivatives and Indefinite Integrals

12.2 Integration by Substitution

12.3 Differential Equations Growth and Decay

12.4 The Definite Integral

12.5 The Fundamental Theorem of Calculus

Chapter 12 Summary and Review

**13. Additional Integration Topics**

13.2 Applications in Business and Economics

13.4 Other Integration Methods

Chapter 13 Summary and Review

**14. Multivariable Calculus**

14.1 Functions of Several Variables

14.4 Maxima and Minima Using Lagrange Multipliers

14.5 Method of Least Squares

14.6 Double Integrals over Rectangular Regions

14.7 Double Integrals over More General Regions

Chapter 14 Summary and Review

**15. Markov Chains (online at** **goo.gl/8SZkyn****)**

15.1 Properties of Markov Chains

15.2 Regular Markov Chains

15.3 Absorbing Markov Chains

Chapter 15 Summary and Review

**Appendix A: Basic Algebra Review**

A.2 Operations on Polynomials

A.4 Operations on Rational Expressions

A.5 Integer Exponents and Scientific Notation

A.6 Rational Exponents and Radicals

**Appendix B: Special Topics (online at goo.gl/mjbXrG)**

B.1 Sequences, Series, and Summation Notation

B.2 Arithmetic and Geometric Sequences

B.4 Interpolating Polynomials and Divided Differences

Table I Integration Formulas

Table II Area under the Standard Normal Curve

## Key Features

- Covers fundamental engineering topics that are presented at the right level, without worry of rigorous proofs
- Includes step-by-step worked examples (of which 100+ feature in the work)
- Provides an emphasis on numerical methods, such as root-finding algorithms, numerical integration, and numerical methods of differential equations
- Balances theory and practice to aid in practical problem-solving in various contexts and applications

## Features

**Personalize learning with MyLab Math.**

MyLab™ Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and engage with media resources to help them absorb course material and understand difficult concepts.

NOTE: This text requires a title-specific MyLab Math access kit** .** The title-specific access kit provides access to the Pirnot

*,*accompanying MyLab course ONLY.

**Mathematics All Around 6/e****Help students overcome math anxiety and develop their skills through applications**

**NEW! Animations**let students interact with the math in a visual, tangible way. These allow students to explore and manipulate the mathematical concepts, leading to more durable understanding. Corresponding exercises in MyLab Math make these truly assignable.**NEW! Interactive Concept Videos**provide a brief explanation, and then the video pauses to ask students to try a problem on their own. Incorrect answers are followed by further explanation taking into consideration what may have led to the student selecting that particular wrong answer.**NEW! An updated Video Program**features a modern, approachable presentation of**new example-based videos**to give students support at home, in a lab, or on the go. Problem-solving methods are reinforced throughout the videos, and**assignable video assessment questions**in MyLab Math give instructors insight into students’ understanding of what they’ve watched.**NEW! Student Success modules**help students succeed in college courses and prepare for their future professions.**NEW! Mindset videos and assignable writing prompts**motivate students to think of their brain as a muscle that can grow, developing new abilities - like doing math - rather than thinking of math as a talent they just don’t have.**Flashcards**are available in a modern, mobile-ready format, so students can study and reinforce vocabulary on the go.**Auto-graded exercise**sets allow students to get instant, personalized feedback while providing instructors insight into how students are performing so that they can evolve their classroom accordingly.**Exercise coverage**gives instructors more flexibility when building assignments and offers students more practice opportunities.**Problem-solving exercises**require students to select a strategy and then work through the stepped-out problem.**Vocabulary exercises**ensure that students understand new terms.**Quiz Yourself**exercises mirror the Quiz Yourself problems in each section, to provide hands-on practice that aligns with the examples in the chapter.

**Foster student engagement and peer-to-peer learning****NEW! Group Projects**have been moved from the text to the MyLab Math course and provide opportunities for students to collaborate. Annotations in the Annotated Instructor’s Edition indicate to instructors when a Group Project might be particularly relevant to the topic at hand.

- MyLab Math provides
**Learning Catalytics**—an interactive, student response tool that uses students’ smartphones, tablets, or laptops to engage them in more sophisticated tasks and thinking.**Premade questions**are available (search tag within Learning Catalytics:**PirnotMAA**). With Learning Catalytics, instructors can pose questions, monitor responses to find out where students are struggling, and manage student interactions with automatic grouping. **NEW! StatCrunch**is now integrated into Pirnot’s MyMathLab course to allow students to harness technology to perform complex analysis on data. StatCrunch isa powerful, web-based statistical software that allows users to analyze and share data sets, and generate compelling reports. The vibrant online community in StatCrunch offers tens of thousands of data sets shared by users.

P

**ersonalized support and targeted practice to help all students succeed.****NEW!**An**Integrated Review MyLab Math course**option provides a complete liberal arts course with embedded review of select developmental topics at the chapter level.**Assignments on prerequisite topics are pre-assigned**in this course–students begin with a Skills Check on prerequisite topics needed for that chapter.**Students who prove mastery**can move on to thecontent, while students who need additional review can remediate using resources like the developmental videos and Integrated Review Worksheets.*Mathematics All Around*- This course solution can be used in a
**co-requisite course model**, or simply to help underprepared students master prerequisite skills and concepts.

**Also available with Integrated Review**The Integrated Review MyLab™ Math course can be used in co-requisite courses, or simply to help students who enter College Algebra without a full understanding of prerequisite skills and concepts. Here’s how it works:

Students begin most

chapters by completing a Skills Check assignment to pinpoint which prerequisite developmental topics, if any, they need to review.*Mathematics All Around with Integrated Review, 6th Edition*Students who demonstrate mastery of the review topics will move straight into the

content.*Mathematics All Around with Integrated Review, 6th Edition*Those who require additional review proceed to a personalized review homework assignment that allows them to remediate on the specific prerequisite topics where they need help.

Students can also review the relevant prerequisite concepts using videos and Integrated Review Worksheets in MyLab Math. The Integrated Review Worksheets are also available in printed form.

**0134800176 / 9780134800172****Mathematics All Around with Integrated Review and Worksheets Plus MyLab Math -- Title-Specific Access Card Package, 6/e****New and Updated Pedagogical Features****Pirnot’s patient writing style and approach help students overcome math anxiety, developing skills through realistic applications that can be seen in the world around them.****UPDATED! Objectives**have been rewritten using action verbs that are measurable and consistent with Bloom’s taxonomy giving students a clear idea of what they are going to learn in each section.**Problem solving**is a theme that carries throughout the text. Section 1.1 discusses 13 problem-solving strategies that will help the students attack problems more effectively. These strategies and principles are strongly reinforced throughout the text with frequent problem-solving reminders in the example solutions and also in Problem-Solving boxes that review the methods from Chapter 1.**UPDATED! Current, relevant applications**have been updated to include fresh ideas that are of interest to today’s students. Chapter openers give an overview of the chapter and also introduce realistic situations relevant to students’ lives that motivate the mathematical tools developed in the chapter. Many section openers have also been rewritten to introduce the concepts in a way students find interesting.**UPDATED!**An increased number of**visual explanations**will help students “see” the mathematics they are learning.**UPDATED! Added and revised examples**feature, clarified explanations, simplified computations, and increased diagrams and annotations in every chapter.**Quiz Yourself**are numerous short quizzes in each section that can be used as a break in the flow of the lecture material to encourage active learning and concept reinforcement. A special symbol indicates when students should try a Quiz Yourself.**Looking Deeper**, the last section of each chapter, discusses an enrichment topic -- fuzzy logic, linear programming, fractals, and more -- to offer an interesting concept beyond the main chapter topic.**UPDATED!**focus on situations that are relevant to students’ lives showing them how math exists all around them and getting them to think more critically. Exercise sets in each section expand upon the material in these highlights.*Math in Your Life: Between the Numbers*boxes- In response to student and instructor feedback, additional
point out common mistakes, provide further advice on problem solving, and make connections between different areas of mathematics.*Some Good Advice*boxes **UPDATED! Using Technology boxes**focus on familiar technologies such as calculators and spreadsheets in addition to more modern free, mobile apps. For ease of use, Using Technology boxes are now integrated into the text near examples where the technology is most appropriate.**NEW!**A printed**Workbook**for students provides additional study support with objective summaries, notetaking, worked-out problems, and additional problems for practice. The Workbook is available for download in MyMathLab or as a printed, unbound, three-hole-punched workbook that can be used as the foundation for a course notebook.

**Exercises and end of chapter material have been fully updated to better support students as they review and practice, while also making it easier for instructors to prepare their assignments.****UPDATED! Exercise sets**have undergone a major revision to reduce repetition, enhance variety, and emphasize current applications. Hundreds of exercises are either new or updated.- Students can use ‘Sharpening Your Skills’ exercises to hone their newly acquired skills.
- An increase in ‘Applying What You’ve Learned’ exercises have been increased and the authors researched hundreds of sources to vary, update, and enrich these application-oriented exercises with real, current data.
- ‘Communicating Mathematics’ asks the students to write about the mathematics they are learning, and have been enhanced based on reviewers’ recommendations.
- To encourage students to read the ‘Math In Your Life: Between the Numbers’ boxes, ‘Math In Your Life: Between the Numbers’ exercises, related to the highlights of the same name, are included in their own category so instructors can assign them and thereby encourage students to read these highlights.
- More rigorous exercises are grouped in the ‘Challenge Yourself’ category.

### New to This Edition

**Personalize learning with MyLab Math.**MyLab™ Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and engage with media resources to help them absorb course material and understand difficult concepts.

NOTE: This text requires a title-specific MyLab Math access kit

The title-specific access kit provides access to the Pirnot*.**,*accompanying MyLab course ONLY.**Mathematics All Around 6/e****Help students overcome math anxiety and develop their skills through applications****Animations**let students interact with the math in a visual, tangible way. These allow students to explore and manipulate the mathematical concepts, leading to more durable understanding. Corresponding exercises in MyLab Math make these truly assignable.**Interactive Concept Videos**provide a brief explanation, and then the video pauses to ask students to try a problem on their own. Incorrect answers are followed by further explanation taking into consideration what may have led to the student selecting that particular wrong answer.**An updated Video Program**features a modern, approachable presentation of**new example-based videos**to give students support at home, in a lab, or on the go. Problem-solving methods are reinforced throughout the videos, and**assignable video assessment questions**in MyLab Math give instructors insight into students’ understanding of what they’ve watched.**Student Success modules**help students succeed in college courses and prepare for their future professions.**Mindset videos and assignable writing prompts**motivate students to think of their brain as a muscle that can grow, developing new abilities - like doing math - rather than thinking of math as a talent they just don’t have.

**Foster student engagement and peer-to-peer learning****Group Projects**have been moved from the text to the MyLab Math course and provide opportunities for students to collaborate. Annotations in the Annotated Instructor’s Edition indicate to instructors when a Group Project might be particularly relevant to the topic at hand.

**StatCrunch**is now integrated into Pirnot’s MyMathLab course to allow students to harness technology to perform complex analysis on data. StatCrunch isa powerful, web-based statistical software that allows users to analyze and share data sets, and generate compelling reports. The vibrant online community in StatCrunch offers tens of thousands of data sets shared by users.

P

**ersonalized support and targeted practice to help all students succeed.**- An
**Integrated Review MyLab Math course**option provides a complete liberal arts course with embedded review of select developmental topics at the chapter level.**Assignments on prerequisite topics are pre-assigned**in this course–students begin with a Skills Check on prerequisite topics needed for that chapter.**Students who prove mastery**can move on to thecontent, while students who need additional review can remediate using resources like the developmental videos and Integrated Review Worksheets.*Mathematics All Around*- This course solution can be used in a
**co-requisite course model**, or simply to help underprepared students master prerequisite skills and concepts.

**New and Updated Pedagogical Features****Pirnot’s patient writing style and approach help students overcome math anxiety, developing skills through realistic applications that can be seen in the world around them.****Objectives**have been rewritten using action verbs that are measurable and consistent with Bloom’s taxonomy giving students a clear idea of what they are going to learn in each section.**Current, relevant applications**have been updated to include fresh ideas that are of interest to today’s students. Chapter openers give an overview of the chapter and also introduce realistic situations relevant to students’ lives that motivate the mathematical tools developed in the chapter. Many section openers have also been rewritten to introduce the concepts in a way students find interesting.- An increased number of
**visual explanations**will help students “see” the mathematics they are learning. **Added and revised examples**feature, clarified explanations, simplified computations, and increased diagrams and annotations in every chapter.*Math in Your Life: Between the Numbers***boxes**focus on situations that are relevant to students’ lives showing them how math exists all around them and getting them to think more critically. Exercise sets in each section expand upon the material in these highlights.**Using Technology boxes**focus on familiar technologies such as calculators and spreadsheets in addition to more modern free, mobile apps. For ease of use, Using Technology boxes are now integrated into the text near examples where the technology is most appropriate.- A printed
**Workbook**for students provides additional study support with objective summaries, notetaking, worked-out problems, and additional problems for practice. The Workbook is available for download in MyMathLab or as a printed, unbound, three-hole-punched workbook that can be used as the foundation for a course notebook.

**Exercises and end of chapter material have been fully updated to better support students as they review and practice, while also making it easier for instructors to prepare their assignments.****Exercise sets**have undergone a major revision to reduce repetition, enhance variety, and emphasize current applications. Hundreds of exercises are either new or updated.- Students can use ‘Sharpening Your Skills’ exercises to hone their newly acquired skills.
- An increase in ‘Applying What You’ve Learned’ exercises have been increased and the authors researched hundreds of sources to vary, update, and enrich these application-oriented exercises with real, current data.
- ‘Communicating Mathematics’ asks the students to write about the mathematics they are learning, and have been enhanced based on reviewers’ recommendations.
- To encourage students to read the Math In Your Life: Between the Numbers boxes, ‘Math In Your Life: Between the Numbers’ exercises, related to the highlights of the same name, are included in their own category so instructors can assign them and thereby encourage students to read these highlights.
- More rigorous exercises are grouped in the ‘Challenge Yourself’ category.

## Table of Contents

Optimization Theory is an active area of research with numerous applications many of the books are designed for engineering classes, and thus have an emphasis on problems from such fields. Covering much of the same material, there is less emphasis on coding and detailed applications as the intended audience is more mathematical. There are still several important problems discussed (especially scheduling problems), but there is more emphasis on theory and less on the nuts and bolts of coding. A constant theme of the text is the &ldquowhy&rdquo and the &ldquohow&rdquo in the subject. Why are we able to do a calculation efficiently? How should we look at a problem? Extensive effort is made to motivate the mathematics and isolate how one can apply ideas/perspectives to a variety of problems. As many of the key algorithms in the subject require too much time or detail to analyze in a first course (such as the run-time of the Simplex Algorithm), there are numerous comparisons to simpler algorithms which students have either seen or can quickly learn (such as the Euclidean algorithm) to motivate the type of results on run-time savings.

Undergraduate and graduate students interested in learning and teaching optimization and operation research.

I think that this book offers a vast and useful outline of many mathematical problems arising from the common ground of optimization theory and operations research. Surely it can be useful and of interest to advanced undergraduates and beginning graduate students concerned with applications of mathematics to optimization problems and related fields.

-- Giorgio Giorgi, Mathematical Reviews

I started reading "Mathematics of Optimization: How to do Things Faster" without a significant background in optimization, linear programming, or operations research. Hence, I really did not know what to expect from the book. I was pleasantly surprised to find the book to be so much fun to work through. The writing is upbeat, entertaining and enlightening and the mathematics is varied, interesting, and inspiring. I am really impressed by "Mathematics of Optimization," and I would love to teach a course based on this book just in order to spend more time going through it myself. I think that the book is unique and should be relevant and of interest to advanced undergraduate and beginning graduate students in pure and applied mathematics and some closely related areas.

## Big Ideas Math Book 7th Grade Accelerated Answer Key | Big Ideas Math Answers 7th Grade Accelerated Solutions Pdf

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## Class 10 Mathematics Notes – All Chapters

Now, you are a matric student of part two and you have an idea how a question paper will be and how will you prepare for it. Let’s discuss some important tips for preparing exams of mathematics subject. My first tip is: you should buy a rough register for doing the practice of mathematics solutions because the more practice you do, the more math will be easy for you. Let’s move forward to further tips. You should prepare chapters in a way that the arrangement of chapters is in the textbook. As a result, if you do any topic, you have all the idea of what is done in a question because everything will be gone through from you. And it will be easy to prepare it from our math notes for class 10.

Furthermore, talking about the exercises of a chapter, you must prepare from the start of the chapter. And understand all the topic and formulas, and by applying that formulas or methods you will have to practice their example questions. Finally, move forward to the exercises. Practice every question of an exercise one by one and if you see a question has many similar parts then you can skip some of them and practice only those parts they are unique or a little bit tough for you. You can do all the exercise in this manner. I recommend you should do questions yourself and if you find any difficulty simply skim the solution from our math notes for class 10.

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## Watch the video: Ratio and Proportion-Review-Exercise-14-Question 3 to 6, Math Lecture. (December 2021).