Question

Evaluate the following integral in cylindrical coordinates triple integral 1/(1+x^2+y^2) dzdxdy z=0..2 y=0..square root(1-x^2) x=-1..1
using pi as needed

Answer 1

In cylindrical coordinates, we take

$x=r\cos\theta$

$y=r\sin\theta$

$z=z$

so that $\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz$.

We have

$1+x^2+y^2=1+r^2$

and the integral is

$\displaystyle\int_0^2\int_0^\pi\int_0^1\frac r{1+r^2}\,\mathrm dr\,\mathrm d\theta\,\mathrm dz=\frac{\ln2}2\int_0^2\int_0^\pi\mathrm d\theta\,\mathrm dz=\boxed{\pi\ln2}$