15.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
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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 x−cos y, R=[0,π]×[0,π]
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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 |
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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.
- 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].
- 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
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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.
- 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.
- 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.
- 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].
- Estimate the average value of the function f on R.
a. 0.112 b. fave≃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.
- 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].
- Estimate the average value of the function f on R.
The solid lying under the surface z=√4−y2 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)=√4−y2 by finding the volume of the corresponding solid.
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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 the following exercises, calculate the integrals by interchanging the order of integration.
∫1−1(∫2−2(2x+3y+5)dx) dy
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40.
∫20(∫10(x+2ey+3)dx) dy
∫271(∫21(3√x+3√y)dy) dx
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812+393√2.
∫161(∫81(4√x+23√y)dy) dx
∫ln 3ln 2(∫10ex+ydy) dx
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e−1.
∫20(∫103x+ydy) dx
∫61(∫92√yy2dy) dx
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15−10√29.
∫91(∫24√xy2dy)dx
In the following exercises, evaluate the iterated integrals by choosing the order of integration.
∫π0∫π/20sin(2x)cos(3y)dx dy
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0.
∫π/8π/12∫π/3π/4[cot x+tan(2y)]dx dy
∫e1∫e1[1xsin(ln x)+1ycos(ln y)]dx dy
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(e−1)(1+sin1−cos1)
∫e1∫e1sin(ln x)cos(ln y)xydx dy
∫21∫21(ln yx+x2y+1)dy dx
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34ln(53)+2b ln22−ln 2
∫e1∫21x2ln(x)dy dx
∫√31∫21y arctan(1x)dy dx
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18[(2√3−3)π+6 ln 2].
∫10∫1/20(arcsin x+arcsin y)dy dx
∫10∫20xex+4ydy dx
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14e4(e4−1).
∫21∫10xex−ydy dx
∫e1∫e1(ln y√y+ln x√x)dy dx
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4(e−1)(2−√e).
∫e1∫e1(x ln y√y+y ln x√x)dy dx
∫10∫21(xx2+y2)dy dx
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−π4+ln(54)−12ln 2+arctan 2.
∫10∫21yx+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]
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12.
f(x,y)=x4+2y3, R=[1,2]×[2,3]
f(x,y)=sinh x+sinh y, R=[0,1]×[0,2]
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12(2 cosh 1+cosh 2−3).
f(x,y)=arctan(xy), R=[0,1]×[0,1]
Let f and g be two continuous functions such that 0≤m1≤f(x)≤M1 for any x∈[a,b] and 0≤m2≤g(y)≤M2 for anyy∈[c,d]. Show that the following inequality is true:
m1m2(b−a)(c−d)≤∫ba∫dcf(x)g(y)dydx≤M1M2(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.
1e2≤∬Re−x2−y2 dA≤1, where R=[0,1]×[0,1]
π2144≤∬Rsin x cosy dA≤π248, where R=[π6,π3]×[π6,π3]
0≤∬Re−y cosx dA≤π2, where R=[0,π2]×[0,π2]
0≤∬R(ln x)(ln y)dA≤(e−1)2, where R=[1,e]×[1,e]
Let f and g be two continuous functions such that 0≤m1≤f(x)≤M1 for any x∈[a,b] and 0≤m2≤g(y)≤M2 for any y∈[c,d]. Show that the following inequality is true:
(m1+m2)(b−a)(c−d)≤∫ba∫dc|f(x)+g(y)| dy dx≤(M1+M2)(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.
2e≤∬R(e−x2+e−y2)dA≤2, where R=[0,1]×[0,1]
π236∬R(sin x+cos y)dA≤π2√336, where R=[π6,π3]×[π6,π3]
π2e−π/2≤∬R(cos x+e−y)dA≤π, where R=[0,π2]×[0,π2]
1e≤∬R(e−y−ln x)dA≤2, where R=[0,1]×[0,1]
In the following exercises, the function f is given in terms of double integrals.
- Determine the explicit form of the function f.
- Find the volume of the solid under the surface z=f(x,y) and above the region R.
- Find the average value of the function f on R.
- 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)=∫y0∫x0(xs+yt)ds dt, where (x,y)∈R=[0,1]×[0,1]
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a. f(x,y)=12xy(x2+y2); b. V=∫10∫10f(x,y)dx dy=18; c. fave=18;
d.
[T] f(x,y)=∫x0∫y0[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
∫ba∫dc|f(x)+g(y)|dy dx=(d−c)∫baf(x)dx
+∫ba∫dcg(y)dy dx=(b−a)∫dcg(y)dy+∫dc∫baf(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].
- 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.
- For m = n = 2, find the average value of f over the region R. Round your answer to the nearest hundredths.
- 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}.
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a. For m = n = 2, I = 4e^{-0.5} \approx 2.43 b. f_{ave} = e^{-0.5} \simeq 0.61;
c.[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].
- 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.
- For m = n = 2, find the average value of f over the region R. Round your answer to the nearest hundredths.
- 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.
- Find the volume of the solids S_n under the surfaces z = f_n(x,y) and above the region R.
- 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]
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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.
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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.
Find the average value of the function f(x,y) = 3xy on the region graphed in the previous exercise.
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\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.
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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.
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\frac{\pi}{2}
The region D bounded by x = y^2 - 1 and x = \sqrt{1 - y^2} as given in the following figure.
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.
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\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.
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.
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\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.
The region D bounded by y = 0 and y = x^2 - 1 as given in the following figure.
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.
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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