Question

Find the volume of the solid enclosed by the surfaces x² + y² + z² = a , x² + y² +z² = b , (a<b) and z = (x² + y² )½

Answer 1

The solid - I'll call it R - is best described in spherical coordinates:

$R = \left\{(\rho,\theta,\varphi) \mid \sqrt{a}\le\rho\le\sqrt{b}, 0\le\theta\le2\pi, 0\le\varphi\le\dfrac\pi4\right\}$

Then the volume of R is

$\displaystyle\iiint_R\mathrm dV = \int_0^{\frac\pi4}\int_0^{2\pi}\int_{\sqrt a}^{\sqrt b}\rho^2\sin(\varphi)\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi \\\\\\ \displaystyle = \boxed{\frac{2\pi}3\left(b^{\frac32}-a^{\frac32}\right)\left(1-\dfrac1{\sqrt2}\right)}$