Let T be the solid consisting of the cylinder $x^2+y^2=1$, $z=cos(x^2+y^2)$ and the surface $z=0$, and let $F(x,y,z)=(-xz-2y)i+(x^2-yz)j+(z^2+1)k$
A) What is the center of mass of T?
B) (Irrelevant but it exists). I assume the F(x,y,z) should be used here.
I know the answer is: $(0,0,\frac{\left(2+sin\left(2\right)\right)}{8\cdot sin\left(1\right)})$
Given by: $\frac{\int _0^{2\pi }\:\int _0^1\:\int _0^{cos\left(r^2\right)}\:z\:r\:dr\:d\theta }{M=\int _0^{2\pi }\:\int _0^1\:cos\left(r^2\right)\:r\:dr\:d\theta \:\:}$
I have been sitting for quite a while, trying to solve the problem not using the formula: $z=\frac{\int _{\:}^{\:}\:\int _{\:}\:\int _S\:zf\left(x,y,z\right)dx\:dy\:dz}{\int _{\:}^{\:}\:\int _{\:}\:\int _S\:f\left(x,y,z\right)dx\:dy\:dz}$ because I was not given the $f(x,y,z)$ directly in the problem. I assumed it could be anything and that there must have been another way to solve the problem, including somehow using the vector field given.
Is there another way to solve the problem given the relevant information or should I always assume $f(x,y,z) $is 1 given such problems where $f(x,y,z)$ is not stated.