Evaluate the following integral $$\iiint\limits_E x \,dV$$ where $E$ is a region bounded by the spheres with radii $2$ and $3$ respectively and center in the origin.
I converted the integral into spherical coordinates: $\int_0^{2\pi} \int_0^\pi \int_2^3 {\rho}^3 \cos\theta$$ \, \sin^2\phi \, d\rho \, d\phi \, d\theta$. After all, got the integral to be $0$. Is it right?
Best Answer
It is an odd function integrated over a symmetric region, so the integral is zero.