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 √(a)\le\rho\le√(b), 0\le\theta\le2\pi, 0\le\varphi\le\frac\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-\frac1{\sqrt2}\right)}`