Find the volume of the solid paraboloid of revolution

Question

Could anyone help me find the volume of the solid paraboloid of revolution $$y=10-x^2-z^2, y\geq0$$

Answer 1

It would help to visualize this differently. This is a body of revolution about the $y$-axis. Therefore we can think of this as the function $y(x)=10-x^2$ to be rotated about the $y$-axis. By Pappus's $2^{nd}$ Centroid theorem the volume is given the by the area multiplied by the path length of it centroid upon rotation, or

$$V=2\pi RA$$

For rotation about the $y$-axis$ the centroid is given by

$$R=\frac{1}{A}\int x\cdot y(x)~dx$$

thus

$$ \begin{align} V &=2\pi \int_0^{\sqrt{10}} x(10-x^2)~dx\\ &=2\pi ~\left(5x^2-\frac{x^4}{4}\right)\biggr|_0^{\sqrt{10}}\\ &=50\pi \end{align} $$

This result has been verified numerically.

Answer 2

HINT

Let's proceed by cylindrical coordinates:

$$x = r \cos{\theta}$$ $$z = r \sin{\theta}$$ $$y = y$$

That is

$$\int_{0}^{2\pi}\int_0^{10}\int_0^{\sqrt{10-y}}r \, dr \, dy\, d\theta $$