1
$\begingroup$

What would be the easiest way to calculate the volume of Hyperboloid with inequation from the picture. I don't know how to approach this problem.

enter image description here

1 Answers 1

3

The area of the cross section taken at any fixed $z \in [-1,1] $ is simply $ 2\pi (1+z^2) $ since the area of an ellipse with semi-axis $a$ and $b$ is $\pi ab.$ Thus if we slice the volume up along the $z$ axis at $z$ and $ z+ \delta z$ then for very small $\delta z$ we have that the volume of the slice is approximately

$\delta V \approx 2\pi (1+z^2) \delta z .$ Adding up all the slices estimates the volume - if we take the limit as $\delta z \to 0 $, so that the sum becomes an integral, we get the precise volume:

$ V = \lim_{\delta z \to 0} \sum_{z=-1}^{1} 2\pi (1+z^2) \delta z = \int^1_{-1} 2\pi (1+z^2) dz = \frac{16\pi}{3}.$