Discussion
The volume of a sphere of radius \( a \) by triple integration is
Answer: B
Integrate the spherical volume element over the full ranges:
\[ V \;=\; \int_{0}^{2\pi}\!\!\int_{0}^{\pi}\!\!\int_{0}^{a} r^{2}\sin\phi\,dr\,d\phi\,d\theta \;=\; (2\pi)(2)\left(\frac{a^{3}}{3}\right) \;=\; \frac{4}{3}\pi a^{3}. \]
The three factors are the longitude range, the integral of \( \sin\phi \) over half a turn, and the radial integral. The value \( 4\pi a^{2} \) is the surface area instead
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