Calculate the triple integral $$\iiint_G\sqrt{x^2+y^2+z^2}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ where $G = \left \{ z = x^2+y^2; y = x; x = 1; y = 0; z = 0 \right \}$
I tried to solve this by switching to spherical coordinates.
I'm not sure if I got the bounds of integration right, but here is what I ended up with.
$$\begin{align} 0 &\leqslant \phi \leqslant \frac{\pi}{4} \\ \text{arccot}\left(\frac{1}{\cos(\phi)}\right) &\leqslant \theta \leqslant \frac{\pi}{2}\phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_} \\ \frac{1}{\sin(\theta)\cos(\phi)} &\leqslant r \leqslant \frac{\cos(\theta)}{\sin^2(\theta)} \end{align}$$
Integral: $$\int ^{\frac{\pi}{4}}_{0} \mathrm{d}\phi\int ^{\frac{\pi}{2}}_{\text{arccot}\left(\frac{1}{\cos(\phi)}\right)} \mathrm{d}\theta\int ^{\frac{1}{\sin(\theta)\cos(\phi)}}_{\frac{\cos(\theta)}{\sin^2(\theta)}}r^2\sin(\theta)\,\mathrm{d}r$$
Any help is greatly appreciated! Thank you!