Evaluate the following integral in cylindrical coordinates
$$\int^{1}_{-1}\int^{\sqrt{1-x^2}}_{0}\int^{2}_{0}\dfrac{1}{1+x^2+y^2}dzdydx$$
My try:
I first took the boundaries as $$-1\le x\le1\\0\le y\le\sqrt{1-x^2}\\0\le z\le2$$
and I know the formula that $$D=\{(r,\theta,z):g(\theta)\le r\le h(\theta),\alpha\le\theta\le\beta,G(x,y)\le z\le H(x,y)\}$$$$\int^{}_{}\int^{}_{D}\int^{}_{}f(r,\theta,z)dV=\int^{\beta}_{\alpha}\int^{h(\theta)}_{g(\theta)}\int^{H(r\cos\alpha,r\sin\theta)}_{G(r\cos\theta,r\sin\theta)}f(r,\theta,z)dzdrd\theta$$
But how to apply this formula and change the boundaries of the integrals?
Can anyone please explain this.
Best Answer
First of all, your $z$ goes from $0$ to $2$ so we just have to look at the $xy$ plane. In the $xy$ plane, you have the upper half circle with radius $1$. So $0\le \theta\le \pi$ and $0\le r\le 1$.
So:
$$\int_0^{\pi}\int_0^1\int_0^2\frac{1}{1+r^2}rdzdrd\theta$$
This integral should be easy enough, use a $u$ sub for the $r$ part.