Chapter 5

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 \(x=1, \quad x=2, \quad y=1,\) and \(y=2,\) and the horizontal plane \(z=0\). $$ f(x, y)=4 x+2 y+8 x y $$

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 \(x=1, \quad x=2, \quad y=1,\) and \(y=2,\) and the horizontal plane \(z=0\). $$ f(x, y)=16 x^{2}+\frac{y}{2} $$

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

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

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 \(\iint_{R} f(x, y) d A\) by using a Riemann sum with \(m=n=2\). Select the sample points to be the upper right corners of the subsquares of \(R\). $$ \begin{array}{|c|c|c|c|} \hline & y_{0}=7 & y_{1}=8 & y_{2}=9 \\ \hline x_{0}=0 & 10.22 & 10.21 & 9.85 \\ \hline x_{1}=1 & 6.73 & 9.75 & 9.63 \\ \hline x_{2}=2 & 5.62 & 7.83 & 8.21 \\ \hline \end{array} $$

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 \(\iint_{R} f(x, y) d A, \quad\) where \(f(x, y)=\sqrt{4-y^{2}},\) by finding the volume of the corresponding solid.

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 \(\iint_{R} f(x, y) d A,\) 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. $$ \int_{-1}^{1}\left(\int_{-2}^{2}(2 x+3 y+5) d x\right) d y $$

In the following exercises, calculate the integrals by interchanging the order of integration. $$ \int_{0}^{2}\left(\int_{0}^{1}\left(x+2 e^{y}-3\right) d x\right) d y $$

In the following exercises, calculate the integrals by interchanging the order of integration. $$ \int_{1}^{27}\left(\int_{1}^{2}(\sqrt[3]{x}+\sqrt[3]{y}) d y\right) d x $$

In the following exercises, calculate the integrals by interchanging the order of integration. $$ \int_{1}^{16}\left(\int_{1}^{8}(\sqrt[4]{x}+2 \sqrt[3]{y}) d y\right) d x $$

In the following exercises, calculate the integrals by interchanging the order of integration. $$ \int_{\ln 2}^{\ln 3}\left(\int_{0}^{1} e^{x+y} d y\right) d x $$

In the following exercises, calculate the integrals by interchanging the order of integration. $$ \int_{1}^{6}\left(\int_{2}^{9} \frac{\sqrt{y}}{x^{2}} d y\right) d x $$

In the following exercises, calculate the integrals by interchanging the order of integration. $$ \int_{1}^{9}\left(\int_{4}^{2} \frac{\sqrt{x}}{y^{2}} d y\right) d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{0}^{\pi} \int_{0}^{\pi / 2} \sin (2 x) \cos (3 y) d x d y $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{\pi / 12}^{\pi / 8} \int_{\pi / 4}^{\pi / 3}[\cot x+\tan (2 y)] d x d y $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{e} \int_{1}^{e}\left[\frac{1}{x} \sin (\ln x)+\frac{1}{y} \cos (\ln y)\right] d x d y $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{e} \int_{1}^{e} \frac{\sin (\ln x) \cos (\ln y)}{x y} d x d y $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{2} \int_{1}^{2}\left(\frac{\ln y}{x}+\frac{x}{2 y+1}\right) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{e} \int_{1}^{2} x^{2} \ln (x) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{\sqrt{3}} \int_{1}^{2} y \arctan \left(\frac{1}{x}\right) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{0}^{1} \int_{0}^{1 / 2}(\arcsin x+\arcsin y) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{0}^{1} \int_{1}^{2} x e^{x+4 y} d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{2} \int_{0}^{1} x e^{x-y} d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{e} \int_{1}^{e}\left(\frac{\ln y}{\sqrt{y}}+\frac{\ln x}{\sqrt{x}}\right) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{1}^{e} \int_{1}^{e}\left(\frac{x \ln y}{\sqrt{y}}+\frac{y \ln x}{\sqrt{x}}\right) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{0}^{1} \int_{1}^{2}\left(\frac{x}{x^{2}+y^{2}}\right) d y d x $$

In the following exercises, evaluate the iterated integrals by choosing the order of integration. $$ \int_{0}^{1} \int_{1}^{2} \frac{y}{x+y^{2}} d y d x $$

In the following exercises, find the average value of the function over the given rectangles. $$ f(x, y)=-x+2 y, \quad R=[0,1] \times[0,1] $$

In the following exercises, find the average value of the function over the given rectangles. $$ f(x, y)=x^{4}+2 y^{3}, \quad R=[1,2] \times[2,3] $$

In the following exercises, find the average value of the function over the given rectangles. $$ f(x, y)=\sinh x+\sinh y, \quad R=[0,1] \times[0,2] $$

In the following exercises, find the average value of the function over the given rectangles. $$ f(x, y)=\arctan (x y), \quad R=[0,1] \times[0,1] $$

Let \(f\) and \(g\) be two continuous functions such that \(0 \leq m_{1} \leq f(x) \leq M_{1} \quad\) for \(\quad\) any \(\quad x \in[a, b] \quad\) and \(0 \leq m_{2} \leq g(y) \leq M_{2}\) for any \(y \in[c, d] .\) Show that the following inequalityis true: \(m_{1} m_{2}(b-a)(c-d) \leq \int_{a}^{b} \int_{c}^{d} f(x) g(y) d y d x \leq M_{1} M_{2}(b-a)(c-d)\).

Let \(f\) and \(g\) be two continuous functions such that \(0 \leq m_{1} \leq f(x) \leq M_{1} \quad\) for \(\quad\) any \(\quad x \in[a, b] \quad\) and \(0 \leq m_{2} \leq g(y) \leq M_{2}\) for any \(y \in[c, d] .\) Show that the following inequality is true: \(\left(m_{1}+m_{2}\right)(b-a)(c-d) \leq \int_{a}^{b} \int_{c}^{d}[f(x)+g(y)] d y d x \leq\left(M_{1}+M_{2}\right)(b-a)(c-d)\).

In the following exercises, the function \(f\) is given in terms of double integrals. a. Determine the explicit form of the function \(f\). b. Find the volume of the solid under the surface \(z=f(x, y)\) and above the region \(R\). c. Find the average value of the function \(f\) on \(R\). d. Use a computer algebra system (CAS) to plot \(z=f(x, y)\) and \(z=f_{\text {ave }}\) in the same system of coordinates. $$ \begin{aligned} &\underline{\phantom{xxx}} \quad \text { [T] } f(x, y)=\int_{0}^{x} \int_{0}^{y}[\cos (s)+\cos (t)] d t d s, \text { where }\\\ &(x, y) \in R=[0,3] \times[0,3] \end{aligned} $$

Show that $$ \int_{a}^{b} \int_{c}^{d} y f(x)+x g(y) d y d x=\frac{1}{2}\left(d^{2}-c^{2}\right)\left(\int_{a}^{b} f(x) d x\right)+\frac{1}{2}\left(b^{2}-a^{2}\right)\left(\int_{c}^{d} g(y) d y\right) $$

[T] Consider the function \(f(x, y)=e^{-x^{2}-y^{2}}\), where \((x, y) \in R=[-1,1] \times[-1,1]\) . a. Use the midpoint rule with \(m=n=2,4, \ldots, 10\) to estimate the double integral \(I=\iint_{R} e^{-x^{2}-y^{2}} d A .\) Round your answers to the nearest hundredths. b. For \(m=n=2\), find the average value of \(f\) over the region \(R\). Round your answer to the nearest hundredths. c. Use a CAS to graph in the same coordinate system the solid whose volume is given by \(\iint_{R} e^{-x^{2}-y^{2}} d A\) and the plane \(z=f_{\text {ave }}\).

[T] Consider the function \(f(x, y)=\sin \left(x^{2}\right) \cos \left(y^{2}\right)\), where \((x, y) \in R=[-1,1] \times[-1,1]\) a. Use the midpoint rule with \(m=n=2,4, \ldots, 10\) to estimate the double integral \(I=\iint_{R} \sin \left(x^{2}\right) \cos \left(y^{2}\right) d A .\) Round your answers to the nearest hundredths. b. For \(m=n=2\), find the average value of \(f\) over the region \(R\). Round your answer to the nearest hundredths. c. Use a CAS to graph in the same coordinate system the solid whose volume is given by \(\iint_{R} \sin \left(x^{2}\right) \cos \left(y^{2}\right) d A\) and the plane \(z=f_{\text {ave }}\).

In the following exercises, the functions \(f_{n}\) are given, where \(n \geq 1\) is a natural number. a. Find the volume of the solids \(S_{n}\) under the surfaces \(z=f_{n}(x, y)\) and above the region \(R\) b. Determine the limit of the volumes of the solids \(S_{n}\) as \(n\) increases without bound. $$ f(x, y)=\frac{1}{x^{n}}+\frac{1}{y^{n}},(x, y) \in R=[1,2] \times[1,2] $$

Show that the average value of a function \(f\) on a rectangular \(\quad\) region \(R=[a, b] \times[c, d]\) is \(f_{\text {ave }} \approx \frac{1}{m n} \sum_{i=1}^{m} \sum_{j=1}^{n} f\left(x_{i j}^{*}, y_{i j}^{*}\right),\) where \(\left(x_{i j}^{*}, y_{i j}^{*}\right)\) 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 $$ f_{\text {ave }} \approx \frac{1}{n^{2}} \sum_{i, j=1}^{n} f\left(\frac{1}{2}\left(x_{i-1}+x_{i}\right), \frac{1}{2}\left(y_{j-1}+y_{j}\right)\right). $$

Specify whether the region is of Type I or Type II. Let \(D\) be the region bounded by the curves of equations \(y=x, y=-x,\) and \(y=2-x^{2}\). Explain why \(D\) is neither of Type I nor II.

Specify whether the region is of Type I or Type II. Let \(D\) be the region bounded by the curves of equations \(y=\cos x\) and \(y=4-x^{2}\) and the \(x\) -axis. Explain why \(D\) is neither of Type I nor II.

Evaluate the double integral \(\iint_{D} f(x, y) d A\) over the region \(D\). \(f(x, y)=2 x+5 y\) and \(D=\left\\{(x, y) \mid 0 \leq x \leq 1, x^{3} \leq y \leq x^{3}+1\right\\}\)

Evaluate the double integral \(\iint_{D} f(x, y) d A\) over the region \(D\). \( f(x, y)=1\) and \(D=\left\\{(x, y) \mid 0 \leq x \leq \frac{\pi}{2}, \sin x \leq y \leq 1+\sin x\right\\}\)

Evaluate the double integral \(\iint_{D} f(x, y) d A\) over the region \(D\). \( f(x, y)=2\) and \(D=\\{(x, y) \mid 0 \leq y \leq 1, y-1 \leq x \leq \arccos y\\}\)

Evaluate the double integral \(\iint_{D} f(x, y) d A\) over the region \(D\). \( f(x, y)=x y\) and \(D=\left\\{(x, y) \mid-1 \leq y \leq 1, y^{2}-1 \leq x \leq \sqrt{1-y^{2}}\right\\}\)

Evaluate the double integral \(\iint_{D} f(x, y) d A\) over the region \(D\). \(f(x, y)=\sin y\) and \(D\) is the triangular region with vertices \((0,0),(0,3),\) and (3,0)

Evaluate the double integral \(\iint_{D} f(x, y) d A\) over the region \(D\). \(f(x, y)=-x+1\) and \(D\) is the triangular region with vertices \((0,0),(0,2),\) and (2,2)

Evaluate the iterated integrals. $$ \int_{0}^{1} \int_{2 x}^{3 x}\left(x+y^{2}\right) d y d x $$