3.7 Triple Integrals in Spherical Coordinates

(a) $\rho=2$, $\theta=\pi$, $\varphi=\frac{\pi}{2}$

(b) $\rho=3$, $\theta=\frac{\pi}{2}$, $\varphi=\frac{\pi}{2}$

(c) $\rho=4$, $\theta=\text{arbitrary}$, $\varphi=\pi$

(d) $\rho=2$, $\theta=\frac{3\pi}{4}$, $\varphi=\frac{\pi}{6}$

(a) $\left(\frac{1}{4}\,,\,\frac{\sqrt{3}}{4}\,,\,\frac{\sqrt{3}}{2}\right)$ (b) $(0,2,0)$

(a) $\varphi=\frac{\pi}{6}\text{ or }\frac{5\pi}{6}$ (b) $\rho=2\cos\varphi$ (c) $\rho\sin\varphi=2$

See the solution.

(a)

(b) $\frac{11\pi}{6}$

$2\pi$

(a) $2\pi\frac{a^3}{3}\big(1-\frac{1}{\sqrt{2}}\big)$ (b) $\frac{\pi a^4}{16}$ (c) $4\pi A\big(a-\sqrt{B}\tan^{-1}\frac{a}{\sqrt{B}}\big)$ (d) $\frac{8}{3}\pi a^3$

(a) $\frac{38}{3}\pi D$

(b) $\bar x = \bar y=0$ $\bar z =\frac{195}{152} \approx 1.28$

(a)

(b) $I=\int_0^{\pi/2}\dee{\varphi}\int_0^{\pi/2}\dee{\theta}\int_0^1\dee{\rho}\ \rho^4\sin^2\varphi\cos\varphi\ \cos\theta$

(c) $\frac{1}{15}$

$\frac{81}{5}$

$\frac{2\pi^2}{3}$

$\frac{64\pi}{9}$

$\pi a^2h$

$\frac{64}{15}\pi\big(1-\frac{1}{2\sqrt{2}}\big)\approx 8.665$

(a) $\frac{\pi}{16}$ (b) The centroid is $(\bar x,\bar y,\bar z)$ with $\bar x=\bar y=0$ and $\bar z=\frac{3}{8}$.

(a) $\dst 9\int_0^2d\rho\int_0^{\pi/2}d\phi\int_0^{2\pi}d\theta\ \rho^3\,\sin\phi\,\cos\phi$ (b) $\frac{1}{2}$

Switch to spherical coordinates.

It does not exist.

(a) $\text{Mass} = \int_0^2\dee{z}\int_0^{2\pi}\dee{\theta}\int_z^2\dee{r}\ r^2$

(b) $\text{Mass} = \int_{\pi/4}^{\pi/2}\dee{\varphi}\int_0^{2\pi}\dee{\theta} \int_0^{2/\sin\varphi}\dee{\rho}\ \rho^3\sin^2\varphi$

(c) $8\pi$

(a) $\int_0^1\dee{r}\int_0^{2\pi}\dee{\theta} \int_r^{\sqrt{2-r^2}}\dee{z}\ r^3$

(b) $\int_0^{\sqrt{2}}\dee{\rho}\int_0^{2\pi}\dee{\theta} \int_0^{\pi/4}\dee{\varphi}\ \rho^4\sin^3\varphi$

(c) $\pi \left[\frac{16\sqrt{2}}{15} - \frac{4}{3}\right] \approx 0.5503$

(a) $\int_0^{\pi/2}\dee{\varphi}\int_0^{\pi/2}\dee{\theta}\int_0^1\dee{\rho}\ \rho^4\sin^2\varphi\cos\varphi\ \cos\theta$ (b) $\int_0^1\dee{z}\int_0^{\pi/2}\dee{\theta}\int_0^{\sqrt{1-z^2}}\dee{r} \ r^2\,z\,\cos\theta$

(c) $\frac{1}{15}$

(a) $\int_0^{3}\dee{\rho}\int_0^{2\pi}\dee{\theta} \int_0^{\pi/6}\dee{\varphi}\ \rho^4\sin^3\varphi$

(b) $\int_0^{3/2}\dee{r}\int_0^{2\pi}\dee{\theta} \int_{\sqrt{3}\,r}^{\sqrt{9-r^2}}\dee{z}\ \ r^3$

(c) $81\pi \left[\frac{4}{5} - \frac{9\sqrt{3}}{20}\right] \approx 5.24$

(a) $\int_0^{1/\sqrt{2}}\dee{r}\int_0^{2\pi}\dee{\theta} \int_r^{\sqrt{1-r^2}}\dee{z}\ r\sqrt{r^2+z^2}$

(b) $\int_0^{1}\dee{\rho}\int_0^{2\pi}\dee{\theta} \int_0^{\pi/4}\dee{\varphi}\ \rho^3\sin\varphi$

(c) $\frac{\pi}{2}\ \left[1-\frac{1}{\sqrt{2}}\right]$

(a) $\frac{2\pi}{3}\left[\big(12\big)^{3/2}+54\right]$

(b) i. The top part is the part of the snowman's head that is inside the sphere
$x^2+y^2+(z-4)^2 = 4$ and above the cone $z-4 = - \sqrt{\frac{x^2+y^2}{3}}$.

ii. The middle part is the part of the snowman's head and body that is bounded on the top by the cone $z-4 = - \sqrt{\frac{x^2+y^2}{3}}$ and is bounded on the bottom by the cone $z = \sqrt{3(x^2+y^2)}$.

iii. The bottom part is the part of the snowman's body that is inside the sphere
$x^2+y^2+z^2 = 12$ and is below the cone $z = \sqrt{3(x^2+y^2)}$.

(b) it is a solid of revolution.

(a) $\frac{2(8^3)\,\pi}{3}\ \frac{12\,\pi}{32}=128\pi^2$

(b) The surface is a torus (a donut) but with the hole in the centre shrunk to a point. The figure below is a sketch of the part of the surface in the first octant.

(a) $I = \int_0^1\dee{r} \int_0^{2\pi}\dee{\theta} \int_0^r\dee{z}\ r\ z\ \sqrt{r^2+z^2}$ (b) $I = \int_{\pi/4}^{\pi/2}\dee{\varphi} \int_0^{2\pi}\dee{\theta} \int_0^{1/\sin\varphi}\dee{\rho}\ \rho^4\sin\varphi \ \cos\varphi$

(c) $\frac{2(2\sqrt{2}-1)\pi}{15}$

(a) $\int_0^a \dee{z}\int_{\pi}^{3\pi/2}\dee{\theta} \int_0^{\sqrt{a^2-z^2}}\dee{r}\ r\big(r^2+z^2\big)^{2014}$ (b) $\int_0^{\pi/2} \dee{\varphi}\int_{\pi}^{3\pi/2}\dee{\theta} \int_0^a\dee{\rho}\ \rho^{4030}\sin\varphi$

(c) $\frac{a^{4031}\pi}{8062}$

(a) $\int_0^1 \dee{r}\ r \int_0^{2\pi}\dee{\theta} \int_{r^2}^r\dee{z}\ z(r^2+z^2)$ (b) $\int_{\pi/4}^{\pi/2} \dee{\varphi} \int_0^{2\pi}\dee{\theta} \int_0^{\cos\varphi/\sin^2\varphi} \dee{\rho}\ \rho^5\sin\varphi \cos\varphi$

(c) $\frac{3\pi}{40}$

(a) $V = \int_0^1\dee{r} \int_0^{\pi/2}\dee{\theta}\int_r^{1+\sqrt{1-r^2}}\dee{z} \ r$ (b) $V = \int_0^{\pi/4}\dee{\varphi} \int_0^{\pi/2}\dee{\theta} \int_0^{2\cos\varphi}\dee{\rho}\ \rho^2\sin\varphi$

(c) $\frac{\pi}{4}$

(a) $\int_0^{2\pi}\dee{\theta}\int_0^{3/2}\dee{r}\ r\int_{\sqrt{3}\,r}^{\sqrt{9-r^2}}dz\ r^2$

(b) $\int_0^{2\pi}\dee{\theta}\int_0^{\pi/6}\dee{\varphi}\int_0^3\dee{\rho}\ \big(\rho^2\sin\varphi\big) \big(\rho^2\sin^2\varphi\big)$

(c) $2\pi \frac{3^5}{5} \big[\frac{2}{3}-\frac{3\sqrt{3}}{8}\big]$