Loading [MathJax]/jax/element/mml/optable/MathOperators.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

4.E: Multiple Integration (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

15.1: Double Integrals over Rectangular Regions

In the following exercises, 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.

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

[Hide Solution]

27.

f(x,y)=16x2+y2

In the following exercises, 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.

f(x,y)=sin xcos y, R=[0,π]×[0,π]

[Hide Solution]

0.

f(x,y)=cos x+cos y, R=[0,π]×[0,π2]

Use the midpoint rule with m=n=2 to estimate , where the values of the function f on R = [8,10] \times [9,11] are given in the following table.

y
x 9 9.5 10 10.5 11
8 9.8 5 6.7 5 5.6
8.5 9.4 4.5 8 5.4 3.4
9 8.7 4.6 6 5.5 3.4
9.5 6.7 6 4.5 5.4 6.7
10 6.8 6.4 5.5 5.7 6.8

[Hide Solution]

21.3.

The values of the function f on the rectangle R = [0,2] \times [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 10.22 10.21 9.85
x_1 = 1 6.73 9.75 9.63
x_2 = 2 5.62 7.83 8.21

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] \times [0,4].
  2. Find the average depth of the swimming pool.
    y
    x 0 1 2 3 4
    0 1 1.5 2 2.5 3
    1 1 1.5 2 2.5 3
    2 1 1.5 1.5 2.5 3
    3 1 1 1.5 2 2.5
    4 1 1 1 1.5 2

[Hide Solution]

a. 28 ft^3 b. 1.75 ft.

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 0 1 2 3
    0 6 6.5 6.4 6
    1 6.5 7 7.5 6.5
    2 6.5 6.7 6.5 6
    3 6 6.5 5 5.6

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

    A series of curves marked k = negative 1, negative ½, negative ¼, negative 1/8, 0, 1/8, ¼, ½, and 1. The line marked k = 0 serves as an asymptote along the line y = x. The lines originate at (along the y axis) 1, 0.7, 0.5, 0.38, 0, (along the x axis) 0.38, 0.5, 0.7, and 1, with the further out lines curving less dramatically toward the asymptote.

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.

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] \times [0.1,0.5].
  2. Estimate the average value of the function f on R.

    A series of quarter circles drawn in the first quadrant marked k = 1/32, 1/16, 1/8, ¼, ½, ¾, and 1. The quarter circles have radii 0. 17, 0.25, 0.35, 0.5, 0.71, 0.87, and 1, respectively.

The solid lying under the surface z = \sqrt{4 - y^2} and above the rectangular region R = [0,2] \times [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.

A quarter cylinder with center along the x axis and with radius 2. It has height 2 as shown.

[Hide Solution]

2\pi

The solid lying under the plane z = y + 4 and above the rectangular region R = [0,2] \times [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 xyz space, a shape is created with sides given by y = 0, x = 0, y = 4, x = 2, z = 0, and the plane the runs from z = 4 along the y axis to z = 8 along the plane formed by y = 4.

In the following exercises, calculate the integrals by interchanging the order of integration.

\int_{-1}^1\left(\int_{-2}^2 (2x + 3y + 5)dx \right) \space dy

[Hide Solution]

40.

\int_0^2\left(\int_0^1 (x + 2e^y + 3)dx \right) \space dy

\int_1^{27}\left(\int_1^2 (\sqrt[3]{x} + \sqrt[3]{y})dy \right) \space dx

[Hide Solution]

\frac{81}{2} + 39\sqrt[3]{2}.

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

\int_{ln \space 2}^{ln \space 3}\left(\int_0^1 e^{x+y}dy \right) \space dx

[Hide Solution]

e - 1.

\int_0^2\left(\int_0^1 3^{x+y}dy \right) \space dx

\int_1^6\left(\int_2^9 \frac{\sqrt{y}}{y^2}dy \right) \space dx

[Hide Solution]

15 - \frac{10\sqrt{2}}{9}.

\int_1^9 \left(\int_4^2 \frac{\sqrt{x}}{y^2}dy \right) dx

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

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

[Hide Solution]

0.

\int_{\pi/12}^{\pi/8}\int_{\pi/4}^{\pi/3} [cot \space x + tan(2y)]dx \space dy

\int_1^e \int_1^e \left[\frac{1}{x}sin(ln \space x) + \frac{1}{y}cos (ln \space y)\right] dx \space dy

[Hide Solution]

(e − 1)(1 + sin1 − cos1)

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

\int_1^2 \int_1^2 \left(\frac{ln \space y}{x} + \frac{x}{2y + 1}\right) dy \space dx

[Hide solution]

\frac{3}{4}ln \left(\frac{5}{3}\right) + 2b \space ln^2 2 - ln \space 2

\int_1^e \int_1^2 x^2 ln(x) dy \space dx

\int_1^{\sqrt{3}} \int_1^2 y \space arctan \left(\frac{1}{x}\right) dy \space dx

[Hide Solution]

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

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

\int_0^1 \int_0^2 xe^{x+4y} dy \space dx

[Hide Solution]

\frac{1}{4}e^4 (e^4 - 1).

\int_1^2 \int_0^1 xe^{x-y} dy \space dx

\int_1^e \int_1^e \left(\frac{ln \space y}{\sqrt{y}} + \frac{ln \space x}{\sqrt{x}}\right) dy \space dx

[Hide Solution]

4(e - 1)(2 - \sqrt{e}).

\int_1^e \int_1^e \left(\frac{x \space ln \space y}{\sqrt{y}} + \frac{y \space ln \space x}{\sqrt{x}}\right) dy \space dx

\int_0^1 \int_1^2 \left(\frac{x}{x^2 + y^2} \right) dy \space dx

[Hide Solution]

-\frac{\pi}{4} + ln \left(\frac{5}{4}\right) - \frac{1}{2} ln \space 2 + arctan \space 2.

\int_0^1 \int_1^2 \frac{y}{x + y^2} dy \space dx

In the following exercises, find the average value of the function over the given rectangles.

f(x,y) = −x +2y, R = [0,1] \times [0,1]

[Hide Solution]

\frac{1}{2}.

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

f(x,y) = sinh \space x + sinh \space y, R = [0,1] \times [0,2]

[Hide solution]

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

f(x,y) = arctan(xy), R = [0,1] \times [0,1]

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

\frac{1}{e^2} \leq \iint_R e^{-x^2 - y^2} \space dA \leq 1, where R = [0,1] \times [0,1]

\frac{\pi^2}{144} \leq \iint_R sin \space x \space cos y \space dA \leq \frac{\pi^2}{48}, where R = \left[ \frac{\pi}{6}, \frac{\pi}{3}\right] \times \left[ \frac{\pi}{6}, \frac{\pi}{3}\right]

0 \leq \iint_R e^{-y}\space cos x \space dA \leq \frac{\pi}{2}, where R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]

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

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

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

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

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

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

In the following exercises, 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.

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

[Hide Solution]

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.

In xyz space, a plane is formed at z = 1/8, and there is another shape that starts at the origin, increases through the plane in a line roughly running from (1, 0.25, 0.125) to (0.25, 1, 0.125), and then rapidly increases to (1, 1, 1).

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

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

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

+ \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.

Show that \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\right) + \frac{1}{2} (b^2 - a^2) \left(\int_c^d g(y)dy\right).

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

[Hide Solution]

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

c.

In xyz space, a plane is formed at z = 0.61, and there is another shape with maximum roughly at (0, 0, 0.92), which decreases along all the sides to the points (plus or minus 1, plus or minus 1, 0.12).

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

  1. Use the midpoint rule with m = n = 2,4,..., 10 to estimate the double integral I = \iint_R sin \space (x^2) \space cos \space (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 \space (x^2) \space cos \space (y^2) \space dA and the plane z = f_{ave}.

In the following exercises, the functions fnfn 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.

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

[Hide Solution]

a. \frac{2}{n + 1} + \frac{1}{4} b. \frac{1}{4}

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

Show that the average value of a function f on a rectangular region R = [a,b] \times [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.

Use the midpoint rule with m = n to show that the average value of a function f on a rectangular region R = [a,b] \times [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)\right).

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.

A contour map showing surface temperature in degrees Fahrenheit. Given the map, the midpoint rule would give rectangles with values 71, 72, 40, and 43.

[Hide Solution]

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

15.2: Double Integrals over General Regions

In the following exercises, specify whether the region is of Type I or Type II.

The region D bounded by y = x^3, \space y = x^3 + 1, \space x = 0, and x = 1 as given in the following figure.

A region is bounded by y = 1 + x cubed, y = x cubed, x = 0, and x = 1.

Find the average value of the function f(x,y) = 3xy on the region graphed in the previous exercise.

[Hide Solution]

\frac{27}{20}

Find the area of the region D given in the previous exercise.

The region D bounded by y = sin \space x, \space y = 1 + sin \space x, \space x = 0, and x = \frac{\pi}{2} as given in the following figure.

A region is bounded by y = 1 + sin x, y = sin x, x = 0, and x = pi/2.

[Hide Solution]

Type I but not Type II

Find the average value of the function f(x,y) = cos \space x on the region graphed in the previous exercise.

Find the area of the region D given in the previous exercise.

[Hide Solution]

\frac{\pi}{2}

The region D bounded by x = y^2 - 1 and x = \sqrt{1 - y^2} as given in the following figure.

A region is bounded by x = negative 1 + y squared and x = the square root of the quantity (1 minus y squared).

Find the volume of the solid under the graph of the function f(x,y) = xy + 1 and above the region in the figure in the previous exercise.

[Hide Solution]

\frac{1}{6}(8 + 3\pi)

The region D bounded by y = 0, \space x = -10 + y, and x = 10 - y as given in the following figure.

A region is bounded by x = negative 10 + y, x = 10 minus y, and y = 0.

Find the volume of the solid under the graph of the function f(x,y) = x + y and above the region in the figure from the previous exercise.

[Hide Solution]

\frac{1000}{3}

The region D bounded by y = 0, \space x = y - 1, \space x = \frac{\pi}{2} as given in the following figure.

A region is bounded by x = pi/2, y = 0, and x = negative 1 + y.

The region D bounded by y = 0 and y = x^2 - 1 as given in the following figure.

A region is bounded by y = 0 and y = negative 1 + x squared.

Type I and Type II

Let D be the region bounded by the curves of equations y = x, \space y = -x and y = 2 - x^2. Explain why D is neither of Type I nor II.

Let D be the region bounded by the curves of equations y = cos \space x and y = 4 - x^2 and the x-axis. Explain why D is neither of Type I nor II.

[Hide Solution]

The region D is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions g_1(x) and g_2(x). The region


This page titled 4.E: Multiple Integration (Exercises) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?