# Thread: Limits in integral..help!

1. ## Limits in integral..help!

The body K is limited by the areas:
z= x^2 + y^2
z= 2-x^2-y^2

Calculate the integral ∫∫∫ (x^2 + y^2) dxdydz

Now, I should introduce cylindrical coordinates, where upon I get that the body is limited by:

z= r^2
z= 2-r^2

My question is about the limits:

r goes from ??
z goes from 2-r^2 to r^2. correct?
the angle goes from 0 to 2pi

I would really appreciate some help

2. The projection on the $xy$ plane is $x^2+y^2=2-x^2-y^2$ or equivalently $x^2+y^2=1$.

So, the limits are:

$\begin{Bmatrix}0\leq \theta\leq 2\pi\\0\leq r \leq 1\\r^2\leq z \leq 2-r^2\end{matrix}$

Regards.

Fernando Revilla