Solution assignment 08 Area and volume

Calculate the volume of the sphere with center $(0,0,0)$ and radius $R$ .

First we look at the circle in the first quadrant and its graph can be represented by the function:

$y=\sqrt{R^2-x^2}$

We revolve this graph around the $X$ -axis and thus get the volume of the half sphere.
So we get:

$\displaystyle\int_{0}^{R}\pi{y^2}dx=\int_{0}^{R}\pi(R^2-x^2)dx=$

$\displaystyle=\pi[R^2x-\frac{1}{3}x^3]_{0}^{R}=\frac{2}{3}\pi{R^3}$

The volume of the whole sphere is: $\displaystyle\frac{4}{3}\pi{R^3}$