Chapter 14.7, Problem 46E

To determine

To determine: The iterated form of the triple integral, ${\displaystyle \int {\displaystyle \underset{Q}{\int }{\displaystyle \int f\left(x,y,z\right)dV}}}$ in the spherical form.

The iterated form of the triple integral, ${\displaystyle \int {\displaystyle \underset{Q}{\int }{\displaystyle \int f\left(x,y,z\right)dV}}}$ in the spherical form is given by,

${\displaystyle {\int }_{{\theta }_1}^{{\theta }_2}{\displaystyle {\int }_{{\varphi }_1}^{{\varphi }_2}{\displaystyle {\int }_{{\rho }_1}^{{\rho }_2}f\left(\rho \sin \varphi \cos \theta ,\rho \sin \varphi \sin \theta ,\rho \cos \varphi \right){\rho }^2\sin \varphi }d\rho }d\varphi }d\theta$.

Consider the rectangular form of the triple integral, ${\displaystyle \int {\displaystyle \underset{Q}{\int }{\displaystyle \int f\left(x,y,z\right)dV}}}$.

Now, the conversion of rectangular coordinates from rectangular to spherical is given as,

$\begin{array}{l}x=\rho \sin \varphi \cos \theta \\ y=\rho \sin \varphi \sin \theta \\ z=\rho \cos \varphi \end{array}$

In the spherical coordinate system, the simplest spherical block is determined by,

$\left\{\left(\rho ,\theta ,\varphi \right):{\rho }_1\le \rho \le {\rho }_2,{\theta }_1\le \theta \le {\theta }_2,{\varphi }_1\le \varphi \le {\varphi }_2\right\}$ where $\rho \ge 0,{\theta }_2-{\theta }_2\le 2\pi$ and $0\le {\varphi }_1\le {\varphi }_2\le \pi$

And the small volume of the block is given by, $dV={\rho }^2\sin \varphi \times d\rho \times d\varphi \times d\theta$.

Therefore, the iterated form of ${\displaystyle \int {\displaystyle \underset{Q}{\int }{\displaystyle \int f\left(x,y,z\right)dV}}}$ in the spherical form is given by,

${\displaystyle {\int }_{{\theta }_1}^{{\theta }_2}{\displaystyle {\int }_{{\varphi }_1}^{{\varphi }_2}{\displaystyle {\int }_{{\rho }_1}^{{\rho }_2}f\left(\rho \sin \varphi \cos \theta ,\rho \sin \varphi \sin \theta ,\rho \cos \varphi \right){\rho }^2\sin \varphi }d\rho }d\varphi }d\theta$