Math – Exercise 5 (Geometry, Analysis)

Given

The semicircle \(y = \sqrt{r^2-x^2}\) with a radius \(r>0\).

Find

The volume of the sphere generated by rotating the semicicle around the x-axis.

Solution

\(
\begin{array}{l}
V_{semicircle} &= 2 \int_0^{r} \sqrt{r^2-x^2}\ dx \\
V_{sphere} &= 2 \pi \int_0^{r} r^2-x^2\ dx = 2 \pi \left[ r^2x-\frac{x^3}{3} \right]_0^r = 2\pi (r^3-\frac{r^3}{3}) = 2\pi \frac{2r^3}{3} = \frac{4 \pi r^3}{3}
\end{array}
\)

Answer

\(V_{sphere} = \frac{4 \pi r^3}{3}\)