# Determining the Surface Area of a Solid of Revolution

## Key Questions

• If a surface is obtained by rotating about the x-axis from $t = a$ to $b$ the curve of the parametric equation

$\left\{\begin{matrix}x = x \left(t\right) \\ y = y \left(t\right)\end{matrix}\right.$,

then its surface area A can be found by

$A = 2 \pi {\int}_{a}^{b} y \left(t\right) \sqrt{x ' \left(t\right) + y ' \left(t\right)} \mathrm{dt}$

If the same curve is rotated about the y-axis, then

$A = 2 \pi {\int}_{a}^{b} x \left(t\right) \sqrt{x ' \left(t\right) + y ' \left(t\right)} \mathrm{dt}$

I hope that this was helpful.