How can I compute the volume of $$S=\{(x,y,z)\in\mathbb R^3\, |\, x^2+y^2 \le\frac{1}{(1+z)^2}, 0\le z\le 1\}$$ by exclusively using integration? I know that I can use Cavalieri, but I don't understand the solution, a step by step explanation would be much appreciated. The result we are looking for is $\frac{\pi}{2}$.
2026-05-05 20:40:27.1778013627
How to use Cavalieri?
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Use cylindrical polar coordinates to get the volume as the following integral $$ \int_0^{2\pi}\int_{0}^1\int_{0}^{1\over 1+z}rdrdzd\theta =\pi\int_0^1{1\over(1+z)^2}dz={\pi\over2} $$