# Error in thomas' calculus?

## Main Question or Discussion Point

Thomas calculus Chapter 15, section "additional and advanced exercises" exercise 8 says:

"Sphere and cylinder" Find the volume of material cut from the
solid sphere $$r^2+z^2 \leq 9$$ by the cylinder $$r = 3 sin \theta$$ "

the answer's book gives de solution

$$\int _0^{\pi }\int _0^{3\sin \theta }\int _0^{\sqrt{9-r^2}}rdzdrd\theta =9\pi$$

I think that's wrong, since Z should be $$-\sqrt{9-r^2}<z<\sqrt{9-r^2}$$, or we would be considering half the cilinder. However, if we subtitute those limits of integrations, we would get as an answer $$18\pi$$, which can not be correct, since the volume of the sphere is $$36\pi$$, and (by looking at the graph of the region) the volume of the cylinder should be LESS than half the volume of the sphere.