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4.E: Multiple Integration (Exercises)

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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 Rf(x,y)dA, where the values of the function f on R=[8,10]×[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]×[7,9] are given in the following table. Estimate the double integral Rf(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.

y0=7 y1=8 y2=9
x0=0 10.22 10.21 9.85
x1=1 6.73 9.75 9.63
x2=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]×[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 ft3 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 Rf(x,y)dA, where R=[0.2,1]×[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. fave0.175; 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 Rf(x,y)dA, where R=[0.1,0.5]×[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=4y2 and above the rectangular regionR=[0,2]×[0,2] is illustrated in the following graph. Evaluate the double integral Rf(x,y), where f(x,y)=4y2 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]

2π

The solid lying under the plane z=y+4 and above the rectangular region R=[0,2]×[0,4] is illustrated in the following graph. Evaluate the double integral Rf(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.

11(22(2x+3y+5)dx) dy

[Hide Solution]

40.

20(10(x+2ey+3)dx) dy

271(21(3x+3y)dy) dx

[Hide Solution]

812+3932.

161(81(4x+23y)dy) dx

ln 3ln 2(10ex+ydy) dx

[Hide Solution]

e1.

20(103x+ydy) dx

61(92yy2dy) dx

[Hide Solution]

151029.

91(24xy2dy)dx

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

π0π/20sin(2x)cos(3y)dx dy

[Hide Solution]

0.

π/8π/12π/3π/4[cot x+tan(2y)]dx dy

e1e1[1xsin(ln x)+1ycos(ln y)]dx dy

[Hide Solution]

(e1)(1+sin1cos1)

e1e1sin(ln x)cos(ln y)xydx dy

2121(ln yx+x2y+1)dy dx

[Hide solution]

34ln(53)+2b ln22ln 2

e121x2ln(x)dy dx

3121y arctan(1x)dy dx

[Hide Solution]

18[(233)π+6 ln 2].

101/20(arcsin x+arcsin y)dy dx

1020xex+4ydy dx

[Hide Solution]

14e4(e41).

2110xexydy dx

e1e1(ln yy+ln xx)dy dx

[Hide Solution]

4(e1)(2e).

e1e1(x ln yy+y ln xx)dy dx

1021(xx2+y2)dy dx

[Hide Solution]

π4+ln(54)12ln 2+arctan 2.

1021yx+y2dy dx

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

f(x,y)=x+2y, R=[0,1]×[0,1]

[Hide Solution]

12.

f(x,y)=x4+2y3, R=[1,2]×[2,3]

f(x,y)=sinh x+sinh y, R=[0,1]×[0,2]

[Hide solution]

12(2 cosh 1+cosh 23).

f(x,y)=arctan(xy), R=[0,1]×[0,1]

Let f and g be two continuous functions such that 0m1f(x)M1 for any x[a,b] and 0m2g(y)M2 for anyy[c,d]. Show that the following inequality is true:

m1m2(ba)(cd)badcf(x)g(y)dydxM1M2(ba)(cd).

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.

1e2Rex2y2 dA1, where R=[0,1]×[0,1]

π2144Rsin x cosy dAπ248, where R=[π6,π3]×[π6,π3]

0Rey cosx dAπ2, where R=[0,π2]×[0,π2]

0R(ln x)(ln y)dA(e1)2, where R=[1,e]×[1,e]

Let f and g be two continuous functions such that 0m1f(x)M1 for any x[a,b] and 0m2g(y)M2 for any y[c,d]. Show that the following inequality is true:

(m1+m2)(ba)(cd)badc|f(x)+g(y)| dy dx(M1+M2)(ba)(cd).

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.

2eR(ex2+ey2)dA2, where R=[0,1]×[0,1]

π236R(sin x+cos y)dAπ2336, where R=[π6,π3]×[π6,π3]

π2eπ/2R(cos x+ey)dAπ, where R=[0,π2]×[0,π2]

1eR(eyln x)dA2, where R=[0,1]×[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=fave in the same system of coordinates.

[T] f(x,y)=y0x0(xs+yt)ds dt, where (x,y)R=[0,1]×[0,1]

[Hide Solution]

a. f(x,y)=12xy(x2+y2); b. V=1010f(x,y)dx dy=18; c. fave=18;

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)=x0y0[cos (s)+cos (t)]dt ds, where (x,y)R=[0,3]×[0,3]

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

badc|f(x)+g(y)|dy dx=(dc)baf(x)dx

+badcg(y)dy dx=(ba)dcg(y)dy+dcbaf(x)dx 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


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