1. ## Triple integral

$\displaystyle \int\!\!\!\!\int\!\!\!\!\int_V e^{{(x^2+y^2+z^2)}^{\frac{3}{2}}}dxdydz$ where $\displaystyle V=\{(x,y,z)|x^2+y^2+z^2\leq1\}$
I tried switching to spherical coordinates:
$\displaystyle x=r\cos\varphi\sin\theta$
$\displaystyle y=r\sin\varphi\sin\theta$
$\displaystyle z=r\cos\theta$
$\displaystyle dV=r^2 \sin \theta dr d \theta d\varphi$
with $\displaystyle 0\leq r\leq1$ but I'm not sure about the range of $\displaystyle \varphi$ and $\displaystyle \theta$

2. Originally Posted by sillyme
$\displaystyle \int\!\!\!\!\int\!\!\!\!\int_V e^{{(x^2+y^2+z^2)}^{\frac{3}{2}}}dxdydz$ where $\displaystyle V=\{(x,y,z)|x^2+y^2+z^2\leq1\}$
I tried switching to spherical coordinates:
$\displaystyle x=r\cos\varphi\sin\theta$
$\displaystyle y=r\sin\varphi\sin\theta$
$\displaystyle z=r\cos\theta$
$\displaystyle dV=r^2 \sin \theta dr d \theta d\varphi$
with $\displaystyle 0\leq r\leq1$ but I'm not sure about the range of $\displaystyle \varphi$ and $\displaystyle \theta$

$\displaystyle 0\leq\varphi\leq2\pi , 0\leq\theta\leq \pi ,0\leq\rho\leq1$ now it is correct

Edited:- Spec show me mistake that I did

3. Actually, with those variables the limits are $\displaystyle 0\leq \varphi \leq 2\pi,\ 0\leq \theta \leq \pi$

It can be a bit confusing since a lot of sources use different variables for the angles.

4. Usually I use $\displaystyle \phi$ instead of $\displaystyle \theta$ so I didn't look to what he write thanks to Spec