When I try to calculate the volume of a sphere of radius a, I set up the following $\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{a}\rho^2\sin{\phi}\ d\rho\ d\phi\ d\theta$, and from this I get $\frac{2}{3}\pi a^3$, which is a half of the supposed $\frac{4}{3}\pi a^3$. Which is wrong in my calculation?
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$$-\cos \pi+ \cos 0 = 2$$ Not the $1$ you probably used by ignoring the $0$ at the lower limit.
answered Mar 31, 2017 at 21:19