I have to evaluate

$$\int^1\_{-1} \int^{ \sqrt {1-x^2}}\_{-\sqrt {1-x^2}} \int^1\_{-\sqrt{x^2+y^2}} \, dz \, dy \, dx$$

using spherical coordinates.

This is what I have come up with

\begin{align}
& \int^1\_0 \int^{2\pi}\_0 \int^{3\pi/4}\_0 r^2\sin\theta \, d\theta \, d\phi \, dr \\[10pt]
= {} & \int^1\_0 r^2 \, dr \int^{2\pi}\_0 d \phi \int^{3\pi/4}\_0 r^2\sin\theta \, d\theta \\[10pt]
= {} & \frac 1 3 \times 2\pi \times \left[-\cos\theta\vphantom{\frac 1 1}\right]^{3\pi/4}\_0
\end{align}

by a combination of sketching and substituting spherical coordinates. 

After evaluating I obtain this integral to equal 3.57.

where as the first one evaluates to 5.236.

EDIT:
A bit of thought shows me that the above integral gives a spherical volume. We need to restrict $r$

As $x^2 + y^2 = 1 \implies \rho = \csc \theta$

$$\int^{3\pi/4}\_{\pi/4} \int^{2\pi}\_0 \int^{\csc \theta}\_1 r^2\sin\theta \, dr d\phi \, d\theta $$

However

This, yet again, does not give me what I want.