Chapter 13

Two integrals Let \(R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\\}\) a. Evaluate \(\iint_{R} \cos (x \sqrt{y}) d A\) b. Evaluate \(\iint_{R} x^{3} y \cos \left(x^{2} y^{2}\right) d A\)

Meaning of the Jacobian The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point \((u, v)\) to the area of the image of that region near the point \((x, y)\) a. Suppose \(S\) is a rectangle in the \(u v\) -plane with vertices \(O(0,0)\) \(P(\Delta u, 0),(\Delta u, \Delta v),\) and \(Q(0, \Delta v)\) (see figure). The image of \(S\) under the transformation \(x=g(u, v), y=h(u, v)\) is a region \(R\) in the \(x y\) -plane. Let \(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) be the images of O, \(P,\) and \(Q,\) respectively, in the \(x y\) -plane, where \(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) do not all lie on the same line. Explain why the coordinates of \(\boldsymbol{O}^{\prime}, \boldsymbol{P}^{\prime},\) and \(Q^{\prime}\) are \((g(0,0), h(0,0)),(g(\Delta u, 0), h(\Delta u, 0))\) and \((g(0, \Delta v), h(0, \Delta v)),\) respectively. b. Use a Taylor series in both variables to show that $$\begin{array}{l} g(\Delta u, 0) \approx g(0,0)+g_{u}(0,0) \Delta u \\ g(0, \Delta v) \approx g(0,0)+g_{v}(0,0) \Delta v \\ h(\Delta u, 0) \approx h(0,0)+h_{u}(0,0) \Delta u \\ h(0, \Delta v) \approx h(0,0)+h_{v}(0,0) \Delta v \end{array}$$ where \(g_{u}(0,0)\) is \(\frac{\partial x}{\partial u}\) evaluated at \((0,0),\) with similar meanings for \(g_{v}, h_{u},\) and \(h_{v}\) c. Consider the vectors \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\) and the parallelogram, two of whose sides are \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\). Use the cross product to show that the area of the parallelogram is approximately \(|J(u, v)| \Delta u \Delta v\) d. Explain why the ratio of the area of \(R\) to the area of \(S\) is approximately \(|J(u, v)|\)

Changing order of integration If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. The solid above the cone \(z=r\) and below the sphere \(\rho=2,\) for \(z \geq 0,\) in the orders \(d z d r d \theta, d r d z d \theta,\) and \(d \theta d z d r\)

Consider the surface \(z=x^{2}-y^{2}\) a. Find the region in the \(x y\) -plane in polar coordinates for which \(z \geq 0\) b. Let \(R=\\{(r, \theta): 0 \leq r \leq a,-\pi / 4 \leq \theta \leq \pi / 4\\},\) which is a sector of a circle of radius \(a\). Find the volume of the region below the hyperbolic paraboloid and above the region \(R\)

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a right circular cone with height \(h\) and base radius \(r\)

Reverse the order of integration in the following integrals. $$\int_{0}^{1} \int_{0}^{\cos ^{-1} y} f(x, y) d x d y$$

Open and closed boxes Consider the region \(R\) bounded by three pairs of parallel planes: \(a x+b y=0, a x+b y=1\) \(c x+d z=0, c x+d z=1, e y+f z=0,\) and \(e y+f z=1\) where \(a, b, c, d, e,\) and \(f\) are real numbers. For the purposes of evaluating triple integrals, when do these six planes bound a finite region? Carry out the following steps. a. Find three vectors \(\mathbf{n}_{1}, \mathbf{n}_{2},\) and \(\mathbf{n}_{3}\) each of which is normal to one of the three pairs of planes. b. Show that the three normal vectors lie in a plane if their triple scalar product \(\mathbf{n}_{1} \cdot\left(\mathbf{n}_{2} \times \mathbf{n}_{3}\right)\) is zero. c. Show that the three normal vectors lie in a plane if ade \(+b c f=0\) d. Assuming \(\mathbf{n}_{1}, \mathbf{n}_{2},\) and \(\mathbf{n}_{3}\) lie in a plane \(P,\) find a vector \(\mathbf{N}\) that is normal to \(P .\) Explain why a line in the direction of \(\mathbf{N}\) does not intersect any of the six planes and therefore the six planes do not form a bounded region. e. Consider the change of variables \(u=a x+b y, v=c x+d z\) \(w=e y+f z .\) Show that $$J(x, y, z)=\frac{\partial(u, v, w)}{\partial(x, y, z)}=-a d e-b c f$$ What is the value of the Jacobian if \(R\) is unbounded?

Changing order of integration If possible, write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume that \(f\) is continuous on the region. $$\begin{aligned}&\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{4 \sec \varphi} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta \text { in the orders }\\\&d \rho d \theta d \varphi \text { and } d \theta d \rho d \varphi\end{aligned}$$

A cake is shaped like a hemisphere of radius 4 with its base on the \(x y\) -plane. A wedge of the cake is removed by making two slices from the center of the cake outward, perpendicular to the \(x y\) -plane and separated by an angle of \(\varphi\) a. Use a double integral to find the volume of the slice for \(\varphi=\pi / 4 .\) Use geometry to check your answer. b. Now suppose the cake is sliced by a plane perpendicular to the \(x y\) -plane at \(x=a > 0 .\) Let \(D\) be the smaller of the two pieces produced. For what value of \(a\) is the volume of \(D\) equal to the volume in part (a)?

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a tetrahedron whose vertices are located at \((0,0,0),(a, 0,0),(0, b, 0),\) and \((0,0, c)\)

Reverse the order of integration in the following integrals. $$\int_{1}^{e} \int_{0}^{\ln x} f(x, y) d y d x$$

A cylindrical soda can has a radius of \(4 \mathrm{cm}\) and a height of \(12 \mathrm{cm} .\) When the can is full of soda, the center of mass of the contents of the can is \(6 \mathrm{cm}\) above the base on the axis of the can (halfway along the axis of the can). As the can is drained, the center of mass descends for a while. However, when the can is empty (filled only with air), the center of mass is once again \(6 \mathrm{cm}\) above the base on the axis of the can. Find the depth of soda in the can for which the center of mass is at its lowest point. Neglect the mass of the can and assume the density of the soda is \(1 \mathrm{g} / \mathrm{cm}^{3}\) and the density of air is \(0.001 \mathrm{g} / \mathrm{cm}^{3}\)

Improper integrals arise in polar coordinates when the radial coordinate \(r\) becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} f(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} f(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\int_{0}^{\pi / 2} \int_{1}^{\infty} \frac{\cos \theta}{r^{3}} r d r d \theta$$

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of the cap of a sphere of radius \(R\) with height \(h\)

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{1} \int_{y}^{1} e^{x^{2}} d x d y$$

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The solid inside the sphere \(\rho=1\) and below the cone \(\varphi=\pi / 4\) for \(z \geq 0\)

Improper integrals arise in polar coordinates when the radial coordinate \(r\) becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} f(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} f(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; R=\\{(r, \theta): 1 \leq r < \infty, 0 \leq \theta \leq 2 \pi\\}$$

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a truncated cone of height \(h\) whose ends have radii \(r\) and \(R\)

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{\pi} \int_{x}^{\pi} \sin y^{2} d y d x$$

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. That part of the solid cylinder \(r \leq 2\) that lies between the cones \(\varphi=\pi / 3\) and \(\varphi=2 \pi / 3\)

A disk of radius \(r\) is removed from a larger disk of radius \(R\) to form an earring (see figure). Assume the earring is a thin plate of uniform density. a. Find the center of mass of the earring in terms of \(r\) and \(R\) (Hint: Place the origin of a coordinate system either at the center of the large disk or at \(Q\); either way, the earring is symmetric about the \(x\) -axis.) b. Show that the ratio \(R / r\) such that the center of mass lies at the point \(P\) (on the edge of the inner disk) is the golden mean \((1+\sqrt{5}) / 2 \approx 1.618\)

Improper integrals arise in polar coordinates when the radial coordinate \(r\) becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} f(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} f(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} e^{-x^{2}-y^{2}} d A ; R=\\{(r, \theta): 0 \leq r < \infty, 0 \leq \theta \leq \pi / 2\\}$$

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of an ellipsoid with axes of length \(2 a\) \(2 b,\) and \(2 c\)

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{1 / 2} \int_{y^{2}}^{1 / 4} y \cos \left(16 \pi x^{2}\right) d x d y$$

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. That part of the ball \(\rho \leq 2\) that lies between the cones \(\varphi=\pi / 3\) and \(\varphi=2 \pi / 3\)

The occurrence of random events (such as phone calls or e-mail messages) is often idealized using an exponential distribution. If \(\lambda\) is the average rate of occurrence of such an event, assumed to be constant over time, then the average time between occurrences is \(\lambda^{-1}\) (for example, if phone calls arrive at a rate of \(\lambda=2 /\) min, then the mean time between phone calls is \(\lambda^{-1}=\frac{1}{2} \mathrm{min}\) ). The exponential distribution is given by \(f(t)=\lambda e^{-\lambda t},\) for \(0 \leq t<\infty\) a. Suppose you work at a customer service desk and phone calls arrive at an average rate of \(\lambda_{1}=0.8 /\) min (meaning the average time between phone calls is \(1 / 0.8=1.25 \mathrm{min}\) ). The probability that a phone call arrives during the interval \([0, T]\) is \(p(T)=\int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t} d t .\) Find the probability that a phone call arrives during the first 45 s \((0.75\) min) that you work at the desk. b. Now suppose that walk-in customers also arrive at your desk at an average rate of \(\lambda_{2}=0.1 /\) min. The probability that a phone $$p(T)=\int_{0}^{T} \int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t} \lambda_{2} e^{-\lambda_{2} x} d t d s$$ Find the probability that a phone call and a customer arrive during the first 45 s that you work at the desk. c. E-mail messages also arrive at your desk at an average rate of \(\lambda_{3}=0.05 /\) min. The probability that a phone call and a customer and an e-mail message arrive during the interval \([0, T]\) is $$p(T)=\int_{0}^{T} \int_{0}^{T} \int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t} \lambda_{2} e^{-\lambda_{2} s} \lambda_{3} e^{-\lambda_{3} u} d t d s d u$$ Find the probability that a phone call and a customer and an e-mail message arrive during the first 45 s that you work at the desk.

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{x}{y^{5}+1} d y d x$$

The limaçon \(r=b+a \cos \theta\) has an inner loop if \(b a\). a. Find the area of the region bounded by the limaçon \(r=2+\cos \theta\) b. Find the area of the region outside the inner loop and inside the outer loop of the limaçon \(r=1+2 \cos \theta\) c. Find the area of the region inside the inner loop of the limaçon $r=1+2 \cos \theta$

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{\sqrt[3]{\pi}} \int_{y}^{\sqrt[3]{\pi}} x^{4} \cos \left(x^{2} y\right) d x d y$$

The following table gives the density (in units of \(\mathrm{g} / \mathrm{cm}^{2}\) ) at selected points of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method. $$\begin{array}{|c|c|c|c|c|c|} \hline & \boldsymbol{\theta}=\mathbf{0} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{2} & \boldsymbol{\theta}=\boldsymbol{3} \pi / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} \\ \hline \boldsymbol{r}=\mathbf{1} & 2.0 & 2.1 & 2.2 & 2.3 & 2.4 \\ \hline \boldsymbol{r}=\mathbf{2} & 2.5 & 2.7 & 2.9 & 3.1 & 3.3 \\ \hline \boldsymbol{r}=\mathbf{3} & 3.2 & 3.4 & 3.5 & 3.6 & 3.7 \\ \hline \end{array}$$

Let \(f\) be a continuous function on \([0,1] .\) Prove that $$\int_{0}^{1} \int_{x}^{1} \int_{x}^{y} f(x) f(y) f(z) d z d y d x=\frac{1}{6}\left(\int_{0}^{1} f(x) d x\right)^{3}$$

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{2} \int_{0}^{4-x^{2}} \frac{x e^{2 y}}{4-y} d y d x$$

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The wedge cut from the cardioid cylinder \(r=1+\cos \theta\) by the planes \(z=2-x\) and \(z=x-2\)

Suppose the density of a thin plate represented by the region \(R\) is \(\rho(r, \theta)\) (in units of mass per area). The mass of the plate is \(\iint_{R} \rho(r, \theta) d A .\) Find the mass of the thin half annulus \(R=\\{(r, \theta): 1 \leq r \leq 4,0 \leq \theta \leq \pi\\}\) with a density \(\rho(r, \theta)=4+r \sin \theta\)

Find the volume of the following solid regions. The solid above the region \(R=\\{(x, y): 0 \leq x \leq 1\), \(0 \leq y \leq 1-x\\}\) bounded by the paraboloids \(z=x^{2}+y^{2}\) and \(z=2-x^{2}-y^{2},\) and the coordinate planes in the first octant

In Section 10.3 it was shown that the area of a region enclosed by the polar curve \(r=g(\theta)\) and the rays \(\theta=\alpha\) and \(\theta=\beta,\) where \(\beta-\alpha \leq 2 \pi,\) is \(A=\frac{1}{2} \int_{\alpha}^{\beta} r^{2} d \theta .\) Prove this result using the area formula with double integrals.

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. Volume of a drilled hemisphere Find the volume of material remaining in a hemisphere of radius 2 after a cylindrical hole of radius 1 is drilled through the center of the hemisphere perpendicular to its base.

Find the volume of the following solid regions. The solid above the parabolic region \(R=\\{(x, y): 0 \leq x \leq 1\), \(\left.0 \leq y \leq 1-x^{2}\right\\}\) and between the planes \(z=1\) and \(z=2-y\)

An important integral in statistics associated with the normal distribution is \(I=\int_{-\infty}^{\infty} e^{-x^{2}} d x .\) It is evaluated in the following steps. a. Assume that $$\begin{aligned} I^{2} &=\left(\int_{-\infty}^{\infty} e^{-x^{2}} d x\right)\left(\int_{-\infty}^{\infty} e^{-y^{2}} d y\right) \\ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^{2}-y^{2}} d x d y \end{aligned}$$ where we have chosen the variables of integration to be \(x\) and \(y\) and then written the product as an iterated integral. Evaluate this integral in polar coordinates and show that \(I=\sqrt{\pi} .\) Why is the solution \(I=-\sqrt{\pi}\) rejected? b. Evaluate \(\int_{0}^{\infty} e^{-x^{2}} d x, \int_{0}^{\infty} x e^{-x^{2}} d x,\) and \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x\) (using part (a) if needed).

Find the volume of the following solid regions. The solid bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=9\)

For what values of \(p\) does the integral \(\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{p}}\) exist in the following cases? a. \(R=\\{(r, \theta): 1 \leq r < \infty, 0 \leq \theta \leq 2 \pi\\}\) b. \(R=\\{(r, \theta): 0 \leq r \leq 1,0 \leq \theta \leq 2 \pi\\}\)

Integrals in strips Consider the integral $$I=\iint_{R} \frac{d A}{\left(1+x^{2}+y^{2}\right)^{2}}$$ where \(R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq a\\}\) a. Evaluate \(I\) for \(a=1 .\) (Hint: Use polar coordinates.) b. Evaluate \(I\) for arbitrary \(a > 0\) c. Let \(a \rightarrow \infty\) in part (b) to find \(I\) over the infinite strip \(R=\\{(x, y): 0 \leq x \leq 1,0 \leq y < \infty\\}\)

Density distribution A right circular cylinder with height \(8 \mathrm{cm}\) and radius \(2 \mathrm{cm}\) is filled with water. A heated filament running along its axis produces a variable density in the water given by \(\rho(r)=1-0.05 e^{-0.01 r^{2}} \mathrm{g} / \mathrm{cm}^{3}(\rho\) stands for density here, not the radial spherical coordinate). Find the mass of the water in the cylinder. Neglect the volume of the filament.

Find the volume of the following solid regions. The solid above the region \(R=\\{(x, y): 0 \leq x \leq 1\), \(0 \leq y \leq 2-x\\}\) and between the planes \(-4 x-4 y+z=0\) and \(-2 x-y+z=8\)

In polar coordinates an equation of an ellipse with eccentricity \(0 < e < 1\) and semimajor axis \(a\) is \(r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}\) a. Write the integral that gives the area of the ellipse. b. Show that the area of an ellipse is \(\pi a b,\) where \(b^{2}=a^{2}\left(1-e^{2}\right)\)

Find the volume of the following solid regions. The solid \(S\) between the surfaces \(z=e^{x-y}\) and \(z=-e^{x-y},\) where \(S\) intersects the \(x y\) -plane in the region \(R=\\{(x, y): 0 \leq x \leq y\), \(0 \leq y \leq 1\\}\)

Gravitational field due to spherical shell A point mass \(m\) is a distance \(d\) from the center of a thin spherical shell of mass \(M\) and radius \(R .\) The magnitude of the gravitational force on the point mass is given by the integral $$F(d)=\frac{G M m}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{(d-R \cos \varphi) \sin \varphi}{\left(R^{2}+d^{2}-2 R d \cos \varphi\right)^{3 / 2}} d \varphi d \theta$$ where \(G\) is the gravitational constant. a. Use the change of variable \(x=\cos \varphi\) to evaluate the integral and show that if \(d>R,\) then \(F(d)=\frac{G M m}{d^{2}},\) which means the force is the same as if the mass of the shell were concentrated at its center. b. Show that if \(d

Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola \(y=x^{2}\) and the line \(y=4\)

Water in a gas tank Before a gasoline-powered engine is started, water must be drained from the bottom of the fuel tank. Suppose the tank is a right circular cylinder on its side with a length of \(2 \mathrm{ft}\) and a radius of 1 ft. If the water level is 6 in above the lowest part of the tank, determine how much water must be drained from the tank.