Triple Integrals in Spherical Coordinates — Question 1

Question 1 – Solution

Strategy. A first-octant spherical shell has constant bounds in all three spherical variables, and the radial integrand cancels one power of the Jacobian.

Step 1: Convert Using $$x=\rho\sin\phi\cos\theta,\quad y=\rho\sin\phi\sin\theta, \quad z=\rho\cos\phi,$$ the bounds are $$0\le\theta\le\frac\pi 2,\qquad 0\le\phi\le\frac\pi 2, \qquad 1\le\rho\le 3,$$ and $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.

Step 2: Evaluate $$\begin{align*} I&=\int_0^{\pi/2}\int_0^{\pi/2}\int_1^3 \frac 1\rho\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\\ &=\left(\frac\pi 2\right)(1)\left[\frac{\rho^2}{2}\right]_1^3 =\boxed{2\pi}. \end{align*}$$

Verification The shell-octant volume is $\frac 18\frac{4\pi}{3}(3^3-1^3)=13\pi/3$. Hence the average integrand is $I/V=6/13$, which lies between its endpoint values $1/3$ and $1$.