$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 7.5E: Excersies

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Exercise $$\PageIndex{1}$$

In the following exercises, evaluate the triple integrals $\iiint_E f(x,y,z) dV$ over the solid $$E$$.

1. $$f(x,y,z) = z, \space B = \{(x,y,z) | x^2 + y^2 \leq 9, \space x \leq 0, \space y \leq 0, \space 0 \leq z \leq 1\}$$

$$\frac{9\pi}{8}$$

2. $$f(x,y,z) = xz^2, \space B = \{(x,y,z) | x^2 + y^2 \leq 16, \space x \geq 0, \space y \leq 0, \space -1 \leq z \leq 1\}$$

3. $$f(x,y,z) = xy, \space B = \{(x,y,z) | x^2 + y^2 \leq 1, \space x \geq 0, \space x \geq y, \space -1 \leq z \leq 1\}$$

$$\frac{1}{8}$$st.

4. $$f(x,y,z) = x^2 + y^2, \space B = \{(x,y,z) | x^2 + y^2 \leq 4, \space x \geq 0, \space x \leq y, \space 0 \leq z \leq 3\}$$

5. $$f(x,y,z) = e^{\sqrt{x^2+y^2}}, \space B = \{(x,y,z) | 1 \leq x^2 + y^2 \leq 4, \space y \leq 0, \space x \leq y\sqrt{3}, \space 2 \leq z \leq 3 \}$$

$$\frac{\pi e^2}{6}$$

6. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B = \{(x,y,z) | 1 \leq x^2 + y^2 \leq 9, \space y \leq 0, \space 0 \leq z \leq 1\}$$

## Exercise $$\PageIndex{2}$$

1.

a. Let $$B$$ be a cylindrical shell with inner radius $$a$$ outer radius $$b$$, and height $$c$$ where $$0 < a < b$$ and $$c>0$$. Assume that a function $$F$$ defined on $$B$$ can be expressed in cylindrical coordinates as $$F(x,y,z) = f(r) + h(z)$$, where $$f$$ and $$h$$ are differentiable functions. If $\int_a^b \bar{f} (r) dr = 0$ and $$\bar{h}(0) = 0$$, where $$\bar{f}$$ and $$\bar{h}$$ are antiderivatives of $$f$$ and $$h$$, respectively, show that

$\iiint_B F(x,y,z) dV = 2\pi c (b\bar{f} (b) - a \bar{f}(a)) + \pi(b^2 - a^2) \bar{h} (c).$

b. Use the previous result to show that

$\iiint_B \left(z + sin \sqrt{x^2 + y^2}\right) dx \space dy \space dz = 6 \pi^2 ( \pi - 2),$

where $$B$$ is a cylindrical shell with inner radius $$\pi$$ outer radius $$2\pi$$, and height $$2$$.

2.

a. Let $$B$$ be a cylindrical shell with inner radius $$a$$ outer radius $$b$$ and height $$c$$ where $$0 < a < b$$ and $$c > 0$$. Assume that a function $$F$$ defined on $$B$$ can be expressed in cylindrical coordinates as F(x,y,z) = f(r) g(\theta) f(z)\), where $$f, \space g,$$ and $$h$$ are differentiable functions. If $\int_a^b \tilde{f} (r) dr = 0,$ where $$\tilde{f}$$ is an antiderivative of $$f$$, show that

$\iiint_B F (x,y,z)dV = [b\tilde{f}(b) - a\tilde{f}(a)] [\tilde{g}(2\pi) - \tilde{g}(0)] [\tilde{h}(c) - \tilde{h}(0)],$

where $$\tilde{g}$$ and $$\tilde{h}$$ are antiderivatives of $$g$$ and $$h$$, respectively.

b. Use the previous result to show that $\iiint_B z \space sin \sqrt{x^2 + y^2} dx \space dy \space dz = - 12 \pi^2,$ where $$B$$ is a cylindrical shell with inner radius $$\pi$$ outer radius $$2\pi$$, and height $$2$$.

## Exercise $$\PageIndex{3}$$

In the following exercises, the boundaries of the solid $$E$$ are given in cylindrical coordinates.

a. Express the region $$E$$ in cylindrical coordinates.

b. Convert the integral $\iiint_E f(x,y,z) dV$ to cylindrical coordinates.

1. E is bounded by the right circular cylinder $$r = 4 \space sin \space \theta$$, the $$r\theta$$-plane, and the sphere $$r^2 + z^2 = 16$$.

a. $$E = \{(r,\theta,z) | 0 \leq \theta \leq \pi, \space 0 \leq r \leq 4 \space sin \space \theta, \space 0 \leq z \leq \sqrt{16 - r^2}\}$$

b. $\int_0^{\pi} \int_0^{4 \space sin \space \theta} \int_0^{\sqrt{16-r^2}} f(r,\theta, z) r \space dz \space dr \space d\theta$

2. $$E$$ is bounded by the right circular cylinder $$r = cos \space \theta$$, the $$r\theta$$-plane, and the sphere $$r^2 + z^2 = 9$$.

3. $$E$$ is located in the first octant and is bounded by the circular paraboloid $$z = 9 - 3r^2$$, the cylinder $$r = \sqrt{r}$$, and the plane $$r(cos \space \theta + sin \space \theta) = 20 - z$$.

a. $$E = \{(r,\theta,z) |0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq r \leq \sqrt{3}, \space 9 - r^2 \leq z \leq 10 - r(cos \space \theta + sin \space \theta)\}; b. $\int_0^{\pi/2} \int_0^{\sqrt{3}} \int_{9-r^2}^{10-r(cos \space \theta + sin \space \theta)} f(r,\theta,z) r \space dz \space dr \space d\theta$ 4. \(E$$ is located in the first octant outside the circular paraboloid $$z = 10 - 2r^2$$ and inside the cylinder $$r = \sqrt{5}$$ and is bounded also by the planes $$z = 20$$ and $$\theta = \frac{\pi}{4}$$.

## Exercise $$\PageIndex{4}$$

In the following exercises, the function $$f$$ and region $$E$$ are given.

a. Express the region $$E$$ and the function $$f$$ in cylindrical coordinates.

b. Convert the integral $\iiint_B f(x,y,z) dV$ into cylindrical coordinates and evaluate it.

1. $$f(x,y,z) = x^2 + y^2$$, $$E = \{(x,y,z) | 0 \leq x^2 + y^2 \leq 9, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x + 3\}$$

a. $$E = \{(r,\theta,z) | 0 \leq r \leq 3, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq z \leq r \space cos \space \theta + 3\},$$ f(r,\theta,z) = \frac{1}{r \space cos \space \theta + 3};

b. $\int_0^3 \int_0^{\pi/2} \int_0^{r \space cos \space \theta+3} \frac{r}{r \space cos \space \theta + 3} dz \space d\theta \space dr = \frac{9\pi}{4}$

2. $$f(x,y,z) = x^2 + y^2, \space E = \{(x,y,z) |0 \leq x^2 + y^2 \leq 4, \space y \geq 0, \space 0 \leq z \leq 3 - x$$

$$f(x,y,z) = x, \space E = \{(x,y,z) | 1 \leq y^2 + z^2 \leq 9, \space 0 \leq x \leq 1 - y^2 - z^2\}$$

a. $$y = r \space cos \space \theta, \space z = r \space sin \space \theta, \space x = z,\space E = \{(r,\theta,z) | 1 \leq r \leq 3, \space 0 \leq \theta \leq 2\pi, \space 0 \leq z \leq 1 - r^2\}, \space f(r,\theta,z) = z$$;

b. $\int_1^3 \int_0^{2\pi} \int_0^{1-r^2} z r \space dz \space d\theta \space dr = \frac{356 \pi}{3}$

3. $$f(x,y,z) = y, \space E = \{(x,y,z) | 1 \leq x^2 + z^2 \leq 9, \space 0 \leq y \leq 1 - x^2 - z^2 \}$$

## Exercise $$\PageIndex{5}$$

In the following exercises, find the volume of the solid $$E$$ whose boundaries are given in rectangular coordinates.

1. $$E$$ is above the $$xy$$-plane, inside the cylinder $$x^2 + y^2 = 1$$, and below the plane $$z = 1$$.

$$\pi$$

2. $$E$$ is below the plane $$z = 1$$ and inside the paraboloid $$z = x^2 + y^2$$.

3. $$E$$ is bounded by the circular cone $$z = \sqrt{x^2 + y^2}$$ and $$z = 1$$.

$$\frac{\pi}{3}$$

4. $$E$$ is located above the $$xy$$-plane, below $$z = 1$$, outside the one-sheeted hyperboloid $$x^2 + y^2 - z^2 = 1$$, and inside the cylinder $$x^2 + y^2 = 2$$.

5. $$E$$ is located inside the cylinder $$x^2 + y^2 = 1$$ and between the circular paraboloids $$z = 1 - x^2 - y^2$$ and $$z = x^2 + y^2$$.

$$\pi$$

6. $$E$$ is located inside the sphere $$x^2 + y^2 + z^2 = 1$$, above the $$xy$$-plane, and inside the circular cone $$z = \sqrt{x^2 + y^2}$$.

7. $$E$$ is located outside the circular cone $$x^2 + y^2 = (z - 1)^2$$ and between the planes $$z = 0$$ and $$z = 2$$.

$$\frac{4\pi}{3}$$

8. $$E$$ is located outside the circular cone $$z = 1 - \sqrt{x^2 + y^2}$$, above the $$xy$$-plane, below the circular paraboloid, and between the planes $$z = 0$$ and $$z = 2$$.

## Exercise $$\PageIndex{6}$$

1. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $\int_{-\pi/2}^{\pi/2} \int_0^1 \int_{r^2}^r r \space dz \space dr \space d\theta.$ Find the volume $$V$$ of the solid. Round your answer to four decimal places.

$$V = \frac{pi}{12} \approx 0.2618$$

2. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $\int_0^{\pi/2} \int_0^1 \int_{r^4}^r r \space dz \space dr \space d\theta.$ Find the volume $$E$$ of the solid Round your answer to four decimal places.

## Exercise $$\PageIndex{7}$$

1. Convert the integral $\int_0^1 \int_{-\sqrt{1-z^2}}^{\sqrt{1-y^2}} \int_{x^2+y^2}^{\sqrt{x^2+y^2}} xz \space dz \space dx \space dy$ into an integral in cylindrical coordinates.

$\int_0^1 \int_0^{\pi} \int_{r^2}^r zr^2 \space cos \space \theta \space dz \space d\theta \space dr\ 2. Convert the integral \[\int_0^2 \int_0^x \int_0^1 (xy + z) dz \space dx \space dy$ into an integral in cylindrical coordinates.

## Exercise $$\PageIndex{8}$$

In the following exercises, evaluate the triple integral $\iiint_B f(x,y,z)dV$ over the solid $$B$$.

1. $$f(x,y,z) = 1, \space B = \{(x,y,z) | x^2 + y^2 + z^2 \leq 90, \space z \geq 0\}$$

$$180 \pi \sqrt{10}$$

2. $$f(x,y,z) = 1 - \sqrt{x^2 + y^2 + z^2}, \space B = \{(x,y,z) | x^2 + y^2 + z^2 \leq 9, \space y \geq 0, \space z \geq 0\}$$

3. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B$$ is bounded above by the half-sphere $$x^2 + y^2 + z^2 = 9$$ with $$z \geq 0$$ and below by the cone $$2z^2 = x^2 + y^2$$.

$$\frac{81\pi(\pi - 2)}{16}$$

4. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B$$ is bounded above by the half-sphere $$x^2 + y^2 + z^2 = 16$$ with $$z \geq 0$$ and below by the cone $$2z^2 = x^2 + y^2$$.

## Exercise $$\PageIndex{9}$$

Show that if $$F ( \rho,\theta,\varphi) = f(\rho)g(\theta)h(\varphi)$$ is a continuous function on the spherical box $$B = \{(\rho,\theta,\varphi) | a \leq \rho \leq b, \space \alpha \leq \theta \leq \beta, \space \gamma \leq \varphi \leq \psi\}$$, then

$\iiint_B F \space dV = \left(\int_a^b \rho^2 f(\rho) \space dr \right) \left( \int_{\alpha}^{\beta} g (\theta) \space d\theta \right)\left( \int_{\gamma}^{\psi} h (\varphi) \space sin \space \varphi \space d\varphi \right).$

## Exercise $$\PageIndex{10}$$

1.

a. A function $$F$$ is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as $$F(x,y,z) = f(\rho)$$, where $$\rho = \sqrt{x^2 + y^2 + z^2}$$. Show that $\iiint_B F(x,y,z) dV = 2\pi \int_a^b \rho^2 f(\rho) d\rho,$ where $$B$$ is the region between the upper concentric hemispheres of radii $$a$$ and $$b$$ centered at the origin, with $$0 < a < b$$ and $$F$$ a spherical function defined on $$B$$.

b. Use the previous result to show that $\iiint_B (x^2 + y^2 + z^2) \sqrt{x^2 + y^2 + z^2} dV = 21 \pi,$ where $$B = \{(x,y,z) | 1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0\}$$.

2.

a. Let $$B$$ be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where $$0 < a < b$$. Consider F a function defined on B whose form in spherical coordinates $$(\rho,\theta,\varphi)$$ is $$F(x,y,z) = f(\rho)cos \space \varphi$$. Show that if $$g(a) = g(b) = 0$$ and $\int_a^b h (\rho)d\rho = 0,$ then $\iiint_B F(x,y,z)dV = \frac{\pi^2}{4} [ah(a) - bh(b)],$ where $$g$$ is an antiderivative of $$f$$ and $$h$$ is an antiderivative of $$g$$.

b. Use the previous result to show that $\iiint_B = \frac{z \space cos \sqrt{x^2 + y^2 + z^2}}{\sqrt{x^2 + y^2 + z^2}}dV = \frac{3\pi^2}{2},$ where $$B$$ is the region between the upper concentric hemispheres of radii $$\pi$$ and $$2\pi$$ centered at the origin and situated in the first octant.

## Exercise $$\PageIndex{11}$$

In the following exercises, the function $$f$$ and region $$E$$ are given.

a. Express the region $$E$$ and function $$f$$ in cylindrical coordinates.

b. Convert the integral $\iiint_B f(x,y,z)dV$ into cylindrical coordinates and evaluate it.

1. $$f(x,y,z) = z; \space E = \{(x,y,z) | 0 \leq x^2 + y^2 + z^2 \leq 1, \space z \geq 0\}$$

2. $$f(x,y,z) = x + y; \space E = \{(x,y,z) | 1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0, \space y \geq 0\}$$

a. $$f(\rho,\theta, \varphi) = \rho \space sin \space \varphi \space (cos \space \theta + sin \space \theta), \space E = \{(\rho,\theta,\varphi) | 1 \leq \rho \leq 2, \space 0 \leq \theta \leq \pi, \space 0 \leq \varphi \leq \frac{\pi}{2}\}$$;

b. $\int_0^{\pi} \int_0^{\pi/2} \int_1^2 \rho^3 cos \space \varphi \space sin \space \varphi \space d\rho \space d\varphi \space d\theta = \frac{15\pi}{8}$

3. $$f(x,y,z) = 2xy; \space E = \{(x,y,z) | \sqrt{x^2 + y^2} \leq z \leq \sqrt{1 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\}$$

4. $$f(x,y,z) = z; \space E = \{(x,y,z) | x^2 + y^2 + z^2 - 2x \leq 0, \space \sqrt{x^2 + y^2} \leq z\}$$

a. $$f(\rho,\theta,\varphi) = \rho \space cos \space \varphi; \space E = \{(\rho,\theta,\varphi) | 0 \leq \rho \leq 2 \space cos \space \varphi, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq \varphi \leq \frac{\pi}{4}\}$$;

b. $\int_0^{\pi/2} \int_0^{\pi/4} \int_0^{2 \space cos \space \varphi} \rho^3 sin \space \varphi \space cos \space \varphi \space d\rho \space d\varphi \space d\theta = \frac{7\pi}{24}$

## Exercise $$\PageIndex{12}$$

In the following exercises, find the volume of the solid $$E$$ whose boundaries are given in rectangular coordinates.

1. $$E = \{ (x,y,z) | \sqrt{x^2 + y^2} \leq z \leq \sqrt{16 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\}$$

2. $$E = \{ (x,y,z) | x^2 + y^2 + z^2 - 2z \leq 0, \space \sqrt{x^2 + y^2} \leq z\}$$

$$\frac{\pi}{4}$$

3. Use spherical coordinates to find the volume of the solid situated outside the sphere $$\rho = 1$$ and inside the sphere $$\rho = cos \space \varphi$$, with $$\varphi \in [0,\frac{\pi}{2}]$$.

4. Use spherical coordinates to find the volume of the ball $$\rho \leq 3$$ that is situated between the cones $$\varphi = \frac{\pi}{4}$$ and $$\varphi = \frac{\pi}{3}$$.

$$9\pi (\sqrt{2} - 1)$$

5. Convert the integral $\int_{-4}64 \int_{-\sqrt{16-y^2}}^{\sqrt{16-y^2}} \int_{-\sqrt{16-x^2-y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^2 dz \space dy \space dx$ into an integral in spherical coordinates.

$\int_0^{\pi/2} \int_0^{\pi/2} \int_0^4 \rho^6 sin \space \varphi \space d\theta$

6. Convert the integral $\int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{16-x^2-y^2}} dz \space dy \space dx$