# 14.2aE: Double Integrals Part 1 (Exercises)

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

$$27$$

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) $$f(x,y) = \sin x - \cos y$$, $$R = [0, \pi] \times [0, \pi]$$

$$0$$

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

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 $$R = [8,10] \times [9,11]$$ 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
$$21.3$$

6) 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

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 $$R = [0,4] \times [0,4]$$.
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
a. 28 $$\text{ft}^3$$
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.
$$y$$
$$x$$ 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

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 $$R = [0.2,1] \times [0,0.8]$$.
2. Estimate the average value of the function $$f$$ on $$R$$. 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 $$R = [0.1,0.5] \times [0.1,0.5]$$.
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$$R = [0,2] \times [0,2]$$ 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.

$$2\pi$$

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

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

$$40$$

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

15) $$\displaystyle \int_1^{27}\left(\int_1^2 (\sqrt{x} + \sqrt{y})\,dy \right) \space dx$$

$$\frac{81}{2} + 39\sqrt{2}$$

16) $$\displaystyle \int_1^{16}\left(\int_1^8 (\sqrt{x} + 2\sqrt{y})\,dy \right) \space dx$$

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

$$e - 1$$

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

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

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

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

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

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

$$0$$

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

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

$$(e − 1)(1 + \sin 1 − \cos 1)$$

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

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

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

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

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

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

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

29) $$\displaystyle \int_0^1 \int_1^2 xe^{x+4y}\,dy \space dx$$

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

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

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

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

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

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

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

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

$$\frac{1}{2}$$

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

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

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

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

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:

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

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 $$R = [0,1] \times [0,1]$$

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

42) $$0 \leq \iint_R e^{-y}\space \cos x \space dA \leq \frac{\pi}{2}$$, where $$R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]$$

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

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:

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

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 $$R = [0,1] \times [0,1]$$

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

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

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

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 $$(x,y) \in R = [0,1] \times [0,1]$$

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 $$(x,y) \in R = [0,3] \times [0,3]$$

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

53) [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 $$z = f_{ave}$$.

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 $$(x,y \in R = [−1,1] \times [−1,1]$$.

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

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

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

57) 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 $$1 \leq j \leq n$$.

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

$f_{ave} \approx \frac{1}{n^2} \sum_{i,j =1}^n f \left(\frac{1}{2} (x_{i=1} + x_i), \space \frac{1}{2} (y_{j=1} + y_j)\right). \nonumber$

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. $$56.5^{\circ}$$ 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.