Solid of revolution of y = √(1+x²)

Text

The curve $y=\sqrt{1+x^{2}}$ revolves about the $x$-axis between $x=0$ and $x=3$. Find the volume of the solid generated.

Solution

By the disk method $V=\pi\displaystyle\int_{0}^{3}y^{2}\,dx$ with $y^{2}=1+x^{2}$: $V=\pi\int_{0}^{3}\left(1+x^{2}\right)dx=\pi\left[x+\frac{x^{3}}{3}\right]_{0}^{3}=\pi\left(3+9\right)=12\pi\approx37.699.$ $\boxed{V=12\pi\approx37.699}$