Discussion
The volume of the cylinder \( x^{2} + y^{2} \le 4 \) between \( z = 0 \) and \( z = 5 \) by triple integration is
Answer: A
In cylindrical coordinates the three integrals separate:
\[ V \;=\; \int_{0}^{2\pi}\!\!\int_{0}^{2}\!\!\int_{0}^{5} r\,dz\,dr\,d\theta \;=\; (2\pi)\left(\frac{4}{2}\right)(5) \;=\; 20\pi. \]
This is \( \pi r^{2}h = \pi(4)(5) \). Using the radius rather than its square would double the answer to \( 40\pi \)
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