What is the volume of this shape after division by π \pi ?

Calculus Level 3

This problem’s question: {\color{#D61F06}\text{This problem's question:}} What is the volume of this shape after division by π \pi ?

This problem does not actually require calculus or even a calculator! Just, think about what the shape actually is.

The formula for the shape: ∣ x 2 + y 2 ∣ 9 + ∣ z ∣ 2 ≤ 1 \frac{\left| x^2+y^2\right| }{9}+\left| z\right| ^2\leq 1 .


The answer is 12.

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1 solution

It is a prolate spheroid, i. e., a sphere stretched by a factor of 3 in both the x and y directions. The pre-stretch radius is 1. Therefore that pre-stretch volume is 4 3 π \frac43\pi . The stretching multiplies the volume by 9. Therefore, the volume is 3 2 4 3 π 3^2 \frac43 \pi and after the division by π \pi gives an answer of 3 × 4 ⇒ 12 3\times 4\Rightarrow 12 .

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