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Double Integrals Part 1 (Exercises) - Mathematics

Double Integrals Part 1 (Exercises) - Mathematics


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In exercises 1 and 2, use the midpoint rule with (m = 4) and (n = 2) to estimate the volume of the solid bounded by the surface (z = f(x,y)), the vertical planes (x = 1), (x = 2), (y = 1), and (y = 2), and the horizontal plane (x = 0).

1) (f(x,y) = 4x + 2y + 8xy)

Answer:
(27)

2) (f(x,y) = 16x^2 + frac{y}{2})

In exercises 3 and 4, estimate the volume of the solid under the surface (z = f(x,y)) and above the rectangular region R by using a Riemann sum with (m = n = 2) and the sample points to be the lower left corners of the subrectangles of the partition.

3) (f(x,y) = sin x - cos y), (R = [0, pi] imes [0, pi])

Answer:
(0)

4) (f(x,y) = cos x + cos y), (R = [0, pi] imes [0, frac{pi}{2}])

5) Use the midpoint rule with (m = n = 2) to estimate (iint_R f(x,y) ,dA), where the values of the function f on (R = [8,10] imes [9,11]) are given in the following table.

(y)
(x)99.51010.511
89.856.755.6
8.59.44.585.43.4
98.74.665.53.4
9.56.764.55.46.7
106.86.45.55.76.8
Answer:
(21.3)

6) The values of the function (f) on the rectangle (R = [0,2] imes [7,9]) are given in the following table. Estimate the double integral (iint_R f(x,y),dA) by using a Riemann sum with (m = n = 2). Select the sample points to be the upper right corners of the subsquares of R.

(y_0 = 7)(y_1 = 8)(y_2 = 9)
(x_0 = 0)10.2210.219.85
(x_1 = 1)6.739.759.63
(x_2 = 2)5.627.838.21

7) The depth of a children’s 4-ft by 4-ft swimming pool, measured at 1-ft intervals, is given in the following table.

  1. Estimate the volume of water in the swimming pool by using a Riemann sum with (m = n = 2). Select the sample points using the midpoint rule on (R = [0,4] imes [0,4]).
  2. Find the average depth of the swimming pool.
    (y)
    (x)01234
    011.522.53
    111.522.53
    211.51.52.53
    3111.522.5
    41111.52
Answer:
a. 28 ( ext{ft}^3)
b. 1.75 ft.

8) The depth of a 3-ft by 3-ft hole in the ground, measured at 1-ft intervals, is given in the following table.

  1. Estimate the volume of the hole by using a Riemann sum with (m = n = 3) and the sample points to be the upper left corners of the subsquares of (R).
  2. Find the average depth of the hole.
    (y)
    (x)0123
    066.56.46
    16.577.56.5
    26.56.76.56
    366.555.6

9) The level curves (f(x,y) = k) of the function (f) are given in the following graph, where (k) is a constant.

  1. Apply the midpoint rule with (m = n = 2) to estimate the double integral (iint_R f(x,y),dA), where (R = [0.2,1] imes [0,0.8]).
  2. Estimate the average value of the function (f) on (R).

Answer:
a. 0.112
b. (f_{ave} ≃ 0.175); here (f(0.4,0.2) ≃ 0.1), (f(0.2,0.6) ≃− 0.2), (f(0.8,0.2) ≃ 0.6), and (f(0.8,0.6) ≃ 0.2)

10) The level curves (f(x,y) = k) of the function (f) are given in the following graph, where (k) is a constant.

  1. Apply the midpoint rule with (m = n = 2) to estimate the double integral (iint_R f(x,y),dA), where (R = [0.1,0.5] imes [0.1,0.5]).
  2. Estimate the average value of the function f on (R).

11) The solid lying under the surface (z = sqrt{4 - y^2}) and above the rectangular region( R = [0,2] imes [0,2]) is illustrated in the following graph. Evaluate the double integral (iint_Rf(x,y)), where (f(x,y) = sqrt{4 - y^2}) by finding the volume of the corresponding solid.

Answer:
(2pi)

12) The solid lying under the plane (z = y + 4) and above the rectangular region (R = [0,2] imes [0,4]) is illustrated in the following graph. Evaluate the double integral (iint_R f(x,y),dA), where (f(x,y) = y + 4), by finding the volume of the corresponding solid.

In the exercises 13 - 20, calculate the integrals by reversing the order of integration.

13) (displaystyle int_{-1}^1left(int_{-2}^2 (2x + 3y + 5),dx ight) space dy)

Answer:
(40)

14) (displaystyle int_0^2left(int_0^1 (x + 2e^y + 3),dx ight) space dy)

15) (displaystyle int_1^{27}left(int_1^2 (sqrt[3]{x} + sqrt[3]{y}),dy ight) space dx)

Answer:
(frac{81}{2} + 39sqrt[3]{2})

16) (displaystyle int_1^{16}left(int_1^8 (sqrt[4]{x} + 2sqrt[3]{y}),dy ight) space dx)

17) (displaystyle int_{ln 2}^{ln 3}left(int_0^1 e^{x+y},dy ight) space dx)

Answer:
(e - 1)

18) (displaystyle int_0^2left(int_0^1 3^{x+y},dy ight) space dx)

19) (displaystyle int_1^6left(int_2^9 frac{sqrt{y}}{y^2},dy ight) space dx)

Answer:
(15 - frac{10sqrt{2}}{9})

20) (displaystyle int_1^9 left(int_4^2 frac{sqrt{x}}{y^2},dy ight),dx)

In exercises 21 - 34, evaluate the iterated integrals by choosing the order of integration.

21) (displaystyle int_0^{pi} int_0^{pi/2} sin(2x)cos(3y),dx space dy)

Answer:
(0)

22) (displaystyle int_{pi/12}^{pi/8}int_{pi/4}^{pi/3} [cot x + an(2y)],dx space dy)

23) (displaystyle int_1^e int_1^e left[frac{1}{x}sin(ln x) + frac{1}{y}cos (ln y) ight] ,dx space dy)

Answer:
((e − 1)(1 + sin 1 − cos 1))

24) (displaystyle int_1^e int_1^e frac{sin(ln x)cos (ln y)}{xy} ,dx space dy)

25) (displaystyle int_1^2 int_1^2 left(frac{ln y}{x} + frac{x}{2y + 1} ight),dy space dx)

Answer:
(frac{3}{4}ln left(frac{5}{3} ight) + 2b space ln^2 2 - ln 2)

26) (displaystyle int_1^e int_1^2 x^2 ln(x),dy space dx)

27) (displaystyle int_1^{sqrt{3}} int_1^2 y space arctan left(frac{1}{x} ight) ,dy space dx)

Answer:
(frac{1}{8}[(2sqrt{3} - 3) pi + 6 space ln 2])

28) (displaystyle int_0^1 int_0^{1/2} (arcsin x + arcsin y),dy space dx)

29) (displaystyle int_0^1 int_0^2 xe^{x+4y},dy space dx)

Answer:
(frac{1}{4}e^4 (e^4 - 1))

30) (displaystyle int_1^2 int_0^1 xe^{x-y},dy space dx)

31) (displaystyle int_1^e int_1^e left(frac{ln y}{sqrt{y}} + frac{ln x}{sqrt{x}} ight),dy space dx)

Answer:
(4(e - 1)(2 - sqrt{e}))

32) (displaystyle int_1^e int_1^e left(frac{x space ln y}{sqrt{y}} + frac{y space ln x}{sqrt{x}} ight),dy space dx)

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

Answer:
(-frac{pi}{4} + ln left(frac{5}{4} ight) - frac{1}{2} ln 2 + arctan 2)

34) (displaystyle int_0^1 int_1^2 frac{y}{x + y^2},dy space dx)

In exercises 35 - 38, find the average value of the function over the given rectangles.

35)(f(x,y) = −x +2y), (R = [0,1] imes [0,1])

Answer:
(frac{1}{2})

36) (f(x,y) = x^4 + 2y^3), (R = [1,2] imes [2,3])

37) (f(x,y) = sinh x + sinh y), (R = [0,1] imes [0,2])

Answer:
(frac{1}{2}(2 space cosh 1 + cosh 2 - 3)).

38) (f(x,y) = arctan(xy)), (R = [0,1] imes [0,1])

39) Let (f) and (g) be two continuous functions such that (0 leq m_1 leq f(x) leq M_1) for any (x ∈ [a,b]) and (0 leq m_2 leq g(y) leq M_2) for any( y ∈ [c,d]). Show that the following inequality is true:

[m_1m_2(b-a)(c-d) leq int_a^b int_c^d f(x) g(y),dy dx leq M_1M_2 (b-a)(c-d).]

In exercises 40 - 43, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.

40) (frac{1}{e^2} leq iint_R e^{-x^2 - y^2} space dA leq 1), where (R = [0,1] imes [0,1])

41) (frac{pi^2}{144} leq iint_R sin x cos y space dA leq frac{pi^2}{48}), where (R = left[ frac{pi}{6}, frac{pi}{3} ight] imes left[ frac{pi}{6}, frac{pi}{3} ight])

42) (0 leq iint_R e^{-y}space cos x space dA leq frac{pi}{2}), where (R = left[0, frac{pi}{2} ight] imes left[0, frac{pi}{2} ight])

43) (0 leq iint_R (ln x)(ln y) ,dA leq (e - 1)^2), where (R = [1, e] imes [1, e] )

44) Let (f) and (g) be two continuous functions such that (0 leq m_1 leq f(x) leq M_1) for any (x ∈ [a,b]) and (0 leq m_2 leq g(y) leq M_2) for any (y ∈ [c,d]). Show that the following inequality is true:

[(m_1 + m_2) (b - a)(c - d) leq int_a^b int_c^d |f(x) + g(y)| space dy space dx leq (M_1 + M_2)(b - a)(c - d)]

In exercises 45 - 48, use property v. of double integrals and the answer from the preceding exercise to show that the following inequalities are true.

45) (frac{2}{e} leq iint_R (e^{-x^2} + e^{-y^2}) ,dA leq 2), where (R = [0,1] imes [0,1])

46) (frac{pi^2}{36}iint_R (sin x + cos y),dA leq frac{pi^2 sqrt{3}}{36}), where (R = [frac{pi}{6}, frac{pi}{3}] imes [frac{pi}{6}, frac{pi}{3}])

47) (frac{pi}{2}e^{-pi/2} leq iint_R (cos x + e^{-y}),dA leq pi), where (R = [0, frac{pi}{2}] imes [0, frac{pi}{2}])

48) (frac{1}{e} leq iint_R (e^{-y} - ln x) ,dA leq 2), where (R = [0, 1] imes [0, 1])

In exercises 49 - 50, the function (f) is given in terms of double integrals.

  1. Determine the explicit form of the function (f).
  2. Find the volume of the solid under the surface (z = f(x,y)) and above the region (R).
  3. Find the average value of the function (f) on (R).
  4. Use a computer algebra system (CAS) to plot (z = f(x,y)) and (z = f_{ave}) in the same system of coordinates.

49) [T] (f(x,y) = int_0^y int_0^x (xs + yt) ds space dt), where ((x,y) in R = [0,1] imes [0,1])

Answer:

a. (f(x,y) = frac{1}{2} xy (x^2 + y^2));
b. (V = int_0^1 int_0^1 f(x,y),dx space dy = frac{1}{8});
c. (f_{ave} = frac{1}{8});

d.

50) [T] (f(x,y) = int_0^x int_0^y [cos(s) + cos(t)] , dt space ds), where ((x,y) in R = [0,3] imes [0,3])

51) Show that if (f) and (g) are continuous on ([a,b]) and ([c,d]), respectively, then

(displaystyle int_a^b int_c^d |f(x) + g(y)| dy space dx = (d - c) int_a^b f(x),dx)

(displaystyle + int_a^b int_c^d g(y),dy space dx = (b - a) int_c^d g(y),dy + int_c^d int_a^b f(x),dx space dy).

52) Show that (displaystyle int_a^b int_c^d yf(x) + xg(y),dy space dx = frac{1}{2} (d^2 - c^2) left(int_a^b f(x),dx ight) + frac{1}{2} (b^2 - a^2) left(int_c^d g(y),dy ight)).

53) [T] Consider the function (f(x,y) = e^{-x^2-y^2}), where ((x,y) in R = [−1,1] imes [−1,1]).

  1. Use the midpoint rule with (m = n = 2,4,..., 10) to estimate the double integral (I = iint_R e^{-x^2 - y^2} dA). Round your answers to the nearest hundredths.
  2. For (m = n = 2), find the average value of f over the region R. Round your answer to the nearest hundredths.
  3. Use a CAS to graph in the same coordinate system the solid whose volume is given by (iint_R e^{-x^2-y^2} dA) and the plane (z = f_{ave}).
Answer:

a. For (m = n = 2), (I = 4e^{-0.5} approx 2.43)
b. (f_{ave} = e^{-0.5} simeq 0.61);

c.

54) [T] Consider the function (f(x,y) = sin (x^2) space cos (y^2)), where ((x,y in R = [−1,1] imes [−1,1]).

  1. Use the midpoint rule with (m = n = 2,4,..., 10) to estimate the double integral (I = iint_R sin (x^2) cos (y^2) space dA). Round your answers to the nearest hundredths.
  2. For (m = n = 2), find the average value of (f)over the region R. Round your answer to the nearest hundredths.
  3. Use a CAS to graph in the same coordinate system the solid whose volume is given by (iint_R sin(x^2) cos(y^2) space dA) and the plane (z = f_{ave}).

In exercises 55 - 56, the functions (f_n) are given, where (n geq 1) is a natural number.

  1. Find the volume of the solids (S_n) under the surfaces (z = f_n(x,y)) and above the region (R).
  2. Determine the limit of the volumes of the solids (S_n) as (n) increases without bound.

55) (f(x,y) = x^n + y^n + xy, space (x,y) in R = [0,1] imes [0,1])

Answer:
a. (frac{2}{n + 1} + frac{1}{4})
b. (frac{1}{4})

56) (f(x,y) = frac{1}{x^n} + frac{1}{y^n}, space (x,y) in R = [1,2] imes [1,2])

57) Show that the average value of a function (f) on a rectangular region (R = [a,b] imes [c,d]) is (f_{ave} approx frac{1}{mn} sum_{i=1}^m sum_{j=1}^n f(x_{ij}^*,y_{ij}^*)),where ((x_{ij}^*,y_{ij}^*)) are the sample points of the partition of (R), where (1 leq i leq m) and (1 leq j leq n).

58) Use the midpoint rule with (m = n) to show that the average value of a function (f) on a rectangular region (R = [a,b] imes [c,d]) is approximated by

[f_{ave} approx frac{1}{n^2} sum_{i,j =1}^n f left(frac{1}{2} (x_{i=1} + x_i), space frac{1}{2} (y_{j=1} + y_j) ight).]

59) An isotherm map is a chart connecting points having the same temperature at a given time for a given period of time. Use the preceding exercise and apply the midpoint rule with (m = n = 2) to find the average temperature over the region given in the following figure.

Answer:
(56.5^{circ}) F; here (f(x_1^*,y_1^*) = 71, space f(x_2^*, y_1^*) = 72, space f(x_2^*,y_1^*) = 40, space f(x_2^*,y_2^*) = 43), where (x_i^*) and (y_j^*) are the midpoints of the subintervals of the partitions of ([a,b]) and ([c,d]), respectively.

Double Integrals Part 1 (Exercises) - Mathematics

Lecture Description

This video lecture, part of the series Calculus Videos: Integration by Prof. , does not currently have a detailed description and video lecture title. If you have watched this lecture and know what it is about, particularly what Mathematics topics are discussed, please help us by commenting on this video with your suggested description and title. Many thanks from,

- The CosmoLearning Team

Course Index

  1. Summation Notation
  2. The Definite Integral: Understanding the Definition
  3. Approximating a Definite Integral Using Rectangles
  4. Trapezoidal Rule to Approximate a Definite Integral
  5. Simpson's Rule to Approximate a Definite Integral
  6. Simpson's Rule and Error Bounds
  7. Calculating a Definite Integral Using Riemann Sums (Part 1)
  8. Calculating a Definite Integral Using Riemann Sums (Part 2)
  9. Basic Integration Formulas
  10. Basic Antiderivate Examples: Indefinite Integral
  11. More Basic Integration Problems
  12. Basic Definite Integral Example
  13. Indefinite Integral: U-substitution
  14. Definite Integral: U-substitution
  15. More Integration Using U-Substitution (Part 1)
  16. More Integration Using U-Substitution (Part 2)
  17. Integration Involving Inverse Trigonometric Functions
  18. Integration By Parts: Indefinite Integral
  19. Integration By Parts: Definite Integral
  20. Indefinite/Definite Integral Examples
  21. Integration By Parts: Using IBP's Twice
  22. Integration By Parts: A "Loopy" Example
  23. Trigonometric Integrals: Part 1 of 6
  24. Trigonometric Integrals: Part 2 of 6
  25. Trigonometric Integrals: Part 3 of 6
  26. Trigonometric Integrals: Part 4 of 6
  27. Trigonometric Integrals: Part 5 of 6
  28. Trigonometric Integrals: Part 6 of 6
  29. Trigonometric Substitution (Part 1)
  30. Trigonometric Substitution (Part 2)
  31. Trigonometric Substitution (Part 3)
  32. Trigonometric Substitution (Part 4)
  33. Trigonometric Substitution (Part 5)
  34. Partial Fractions: Decomposing a Rational Function
  35. Partial Fractions: Coefficients of a Partial Fraction Decomposition
  36. Partial Fractions: Problem
  37. Partial Fractions: Problem Using a Rationalizing Substitution
  38. Calculating Double Integrals Over Rectangular Regions
  39. Calculating Double Integrals Over General Regions
  40. Reversing the Order of Integration (Part 1)
  41. Reversing the Order of Integration (Part 2)
  42. Finding Areas in Polar Coordinates
  43. Double Integral Using Polar Coordinates (Part 1)
  44. Double Integral Using Polar Coordinates (Part 2)
  45. Double Integral Using Polar Coordinates (Part 3)
  46. Triple Integrals
  47. Triple Integrals in Spherical Coordinates
  48. Line Integrals
  49. Solving First Order Linear Differential Equations
  50. Finding Centroids/Centers of Mass (Part 1)
  51. Finding Centroids/Centers of Mass (Part 2)
  52. Improper Integrals: Introduction
  53. Improper Integrals: Using L'Hospitals Rule
  54. Improper Integrals: Infinity in the Upper and Lower Limit
  55. Improper Integrals: Infinite Discontinuity at an Endpoint
  56. Improper Integrals: Infinite Discontinuity in the Middle of the Interval
  57. Volumes of Revolution: Disk/Washer Method & Rotating Regions About a Horizontal Line
  58. Volumes of Revolution: Disk/Washer Method & Rotating Regions About a Vertical Line
  59. Volumes of Revolution: Disk/Washer Method (cont.)
  60. Work Problems: Finding the Work To Empty a Tank Full of Water

Course Description


In this course, Calculus Instructor Patrick gives 60 video lectures on Integral Calculus. Some of the topics covered are: Indefinite Integrals, Definite Integrals, Trigonometric Integrals, Trigonometric Substitution, Partial Fractions, Double Integrals, Triple Integrals, Polar Coordinates, Spherical Coordinates, Line Integrals, Centroids/Centers of Mass, Improper Integrals, Volumes of Revolution, Work, and many more.


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Welcome!

This is one of over 2,400 courses on OCW. Explore materials for this course in the pages linked along the left.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse (just remember to cite OCW as the source.)

About MIT OpenCourseWare

MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. Learn more »

© 2001&ndash2018
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Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use.


Welcome!

This is one of over 2,400 courses on OCW. Explore materials for this course in the pages linked along the left.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

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MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. Learn more »

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Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use.


Application Of Double Integrals Quiz

Distance travelled by any object is

Double integral of its accelecration

Double integral of its velocity

Double integral of its Force

Double integral of its Momentum

Explanation: We know that,
x(t) = &int&inta(t) dtdt.

Find the distance travelled by a car moving with acceleration given by a(t)=t 2 + t, if it moves from t = 0 sec to t = 10 sec, if velocity of a car at t = 0sec is 40 km/hr.

Explanation: Add constant automatically
We know that,

Find the distance travelled by a car moving with acceleration given by a(t)=Sin(t), if it moves from t = 0 sec to t = &pi/2 sec, if velocity of a car at t=0sec is 10 km/hr.

Explanation: Add constant automatically
We know that,

Find the distance travelled by a car moving with acceleration given by a(t)=t 2 &ndash t, if it moves from t = 0 sec to t = 1 sec, if velocity of a car at t = 0sec is 10 km/hr.


Test: Double And Triple Integrals - 1

The value of dxdy changing the order of integration is

The volume of ellipsoide is

abc cubic units

abc cubic units

The area bounded by the curve y = &psi(x), x-axis and the lines x = l , x = m(l <m ) is given by

The volume of an object expressed in spherical coordinates is given by sin &phi dr d&phi d&theta. The value of the integral is

dx dy is equal to

Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integral is

The area bounded by the parabola y 2 = 4ax and straight line x + y = 3a is

Consider the shaded triangular region P shown in the figure, what is the value of ?

We assume x + 2 = t 2 dx = 2t dt

To evaluate over the region A bounded by the curve r = r1, r = r2 and the straight lines &theta = &theta1, &theta = &theta​2, we first integrate

r between limits r = r1 and r = r2 treating &theta as a constant

&theta between the limits &theta = &theta1 and &theta2 treating r as a constant

To change Cartesian plane (x, y, z) to spherical polar coordinates (r, &theta, &phi)

x = r sin &theta sin &phi, y = r cos&theta cos &phi, z = r cos &theta

x = r sin &theta cos &phi, y = r cos &theta sin &phi, z = r cos &theta

x = r sin &phi cos &phi, y = r sin &theta sin &phi, z = r cos &phi

r 2 = x 2 + y 2 + z 2 , tan&theta = y/x, &phi = arccos(z/&radic(x 2 +y 2 +z 2 ) )

To convert a point from Cartesian coordinates to spherical coordinates, use equations

r 2 = x 2 + y 2 + z 2 , tan&theta = y/x, &phi = arccos(z/&radic(x 2 +y 2 +z 2 ) )

If the triple integral over the region bounded by the planes 2x + y + z = 4, x = 0, y = 0, z = 0 is given by then the function &lambda(x) &ndash &pi(x, y) is

where V is region bounded by the plane 2x + y + z = 4 x = 0, y = 0, z = 0

What is the total mass of cube between the limits 0&le x&le 1, 0&le y&le 1, 0 &lez &le1 at any point given by xyz?

Area bounded by the curves y 2 = x 3 and x 2 = y 3 is

The value of (x+y+z) dzdydx is

By changing the order of integration in the value is

By the change of variable x(u, v) = uv, y(u, v) = u/v is double integral, the integrand f(x, y) change to . Then, &phi(u,v) is

The volume of the tetrahedron bounded by the plane and the co-ordinate planes is equal to

Here
Let u = x/a, v = y/b , w = z/c
Then dx = a du, dy = b dv, dz = c dw
So, Required volume
V = abc du dv dw
where u + v + w &le 1, u ,v ,w &ge 0
Thus


Hence, the correct answer is (d)

The value of integral dxdy is


Math H53: Honors Multivariable Calculus

Instructor Office hours: Regular office hours: 4:30-5:30 on Tuesday and 2-3:30 on Thursday. Check Bcourses "Syllabus" for Zoom ID. Always feel free to send me questions or ask for alternative office hours.

Final exam: Check UC Berkeley final exam schedule

Prerequisites: Math 1B or equivalent.

Text: The primary texts for this course are Vector Calculus by Michael Corral ([Co]) and Notes on Multivariable Calculus by Cain and Herod ([CH]). Students should feel free to consult other books for additional exercises and/or alternative presentations of the material. Wikipedia also has lots of great articles on the topics at hand. Students are expected to read the relevant sections of the notes, as the lectures are meant to complement the notes, not replace it, and we have a lot of material to cover.

Grading: Your homework grade (hw) will be the average of all homeworks, with the lowest dropped. Your exam grade (exams) will be computed based on the maximum of the following three schemes: (0.2)MT1 + (0.2)MT2 + (0.4)F (0.2)MT1 + (0.6)F (0.2)MT2 + (0.6)F. Finally, your total grade will be calculated as the maximum of: (0.2)hw + (0.8)exams, (0.3)hw + (0.7)exams.

Website: For now, the only website is this page, http://math.berkeley.edu/

dcorwin/mathh53s21.html. I will use bcourses for solutions and other non-public information, such as book excerpts, exams, and my phone number.

  • Homework will be assigned regularly (see the syllabus) and due at 11pm on Gradescope. I grant extensions in reasonable circumstances, but you must talk to me as early as possible. The longer you wait, the less flexible I will be.
  • You may work together to figure out homework problems, but you must write up your solutions in your own words in order to receive credit. In particular, please do not copy answers from the internet or solution manuals. Since a major purpose of the homework is to prepare you for the exams, I encourage you to give each problem an honest shot by yourself (say, at least thirty minutes) before discussing it with others. Another useful practice is if you're stuck on a problem, come to office hours and ask for a hint. The more you figure out on your own, the better your understanding of the material, and the better you'll do both on the exams and in your future endeavours that might require abstract algebra.
  • You may cite any results from the notes, unless otherwise stated.
  • The usual expectations and procedures for academic integrity at UC Berkeley apply. Cheating on an exam will result in a failing grade and will be reported to the University Office of Student Conduction. Please don't put me through this.
  • Please let me know sooner rather than later if you need any accommodations related to the Disabled Student Program (DSP). I am more than happy to make arrangements, but it really helps if you tell me earlier rather than later.
  • Per university guidelines, it is your responsibility to notify the instructor in writing by the end of the second week of classes (January 31) of any scheduling conflicts due to religious observance or extracurricular activities, and to propose a resolution for those conflicts.

Additional resources (will be on Bcourses when needed):

  • Calculus: Early Transcendentals by James Stewart, denoted [S]
  • Calculus Volume II by Tom Apostol, denoted [A]
  • div grad curl and all that by H. M. Schey, denoted [dgcaat]
  • Line Integrals and Green's Theorem by Jeremy Orloff, denoted [O]

Course Overview: Outlined below is the rough course schedule. Depending on how the class progresses it may be subject to minor changes over the course of the semester.


About calculating limits of integrals (Part 1)

Say $a, x$ and $y$ are three real number constants and $t$ is a real variable. Now define the complex number, $z = -y + i(a+x-t)$ and consider an integral of the form $int_^ z ext< >tanh (pi z) log (z^2 + a^2) dz$ for some real functions $f$ and $g$.

Obviously $dz = -i dt$ and hence its an integral over $t$. Assume that it is ensures that the range of $t$ over which the integration is being done is such that the integrand doesn't hit any of its poles or branch points/cuts of the integrand.

Now what I want is to calculate $lim_ [int_^ z ext< > anh (pi z) log (z^2 + a^2) dz] $

Clearly this integral isn't doable that I can just do the integral and then take the limit at the end.

Firstly if it is true that $lim_ f(x,y) = lim_ g(x,y) $ then can I immediately conclude that the limiting value of the integral is $ without any further analysis?

Can I say expand the integrand $z ext< > anh(pi z) log(z^2 +a^2)$ in a Taylor series in $x$ and $y$ and then do the integration on the constant term thus obtained and then take the limit? (because any term of the series with a non-zero power of $x$ and/or $y$ will obviously die off in the eventual limit)

Can I also separately take the $x,y ightarrow 0$ limit on the limits of the integrand and thus avoid calculating the full integral with the full $f$ and $g$ ?

Could I have just substituted $z = i(a-t)$ in the integrand from the beginning itself?


Watch the video: Double Integrals (July 2022).


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