Question

(x2 + y2) dv e , where e lies between the spheres x2 + y2 + z2 = 9 and x2 + y2 + z2 = 16.

Answer 1

Convert to spherical coordinates.


`\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi`


`\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_(\varphi=0)^(\varphi=\pi)\int_(\theta=0)^(\theta=2\pi)\int_(\rho=3)^(\rho=4)\underbrace{\rho^2\sin^2\varphi}_(x^2+y^2)\underbrace{\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi}_(\mathrm dV)`

`=\displaystyle2\pi\left(\int_(\varphi=0)^(\varphi=\pi)\sin^3\varphi\,\mathrm d\varphi\right)\left(\int_(\rho=3)^(\rho=4)\rho^4\,\mathrm d\rho\right)=(6248\pi)/(15)`