Evaluate $\int \int \int_B x^2+y^2 \, dxdydz$

Solution 1:

From symmetry of the region, it is best to integrate in cylindrical coordinates $( r, \phi, z ) $

The bounds are $ z = \dfrac{1}{2} r^2 $ and $ z = 2 $

So $ r $ will range from $0$ to $ 2$.

$I = \displaystyle \int_{\phi = 0 }^{2\pi} \int_{r = 0}^{2} \int_{ z = \dfrac{1}{2} r^2 }^{2} dz r^3 dr d\phi $

Integrating with respect to $\phi$ and $z$ simplifies the above to

$I = \displaystyle 2 \pi \int_{r = 0 }^2 \left( 2 r^3 - \dfrac{1}{2} r^5 \right) dr $

And this evaluates to

$ I = \displaystyle 2 \pi \left( \dfrac{1}{2} r^4 - \dfrac{1}{12} r^6 \right)\biggr| _{0}^{2} = \dfrac{16 \pi}{3} $