# Area of Surface of Revolution Calculator

## Calculate the area of a surface of revolution step by step

The calculator will find the area of the surface of revolution (around the given axis) of the explicit, polar, or parametric curve on the given interval, with steps shown.

Choose type:

Enter a function:

Rotate around the -axis

Enter a lower limit:

If you need -oo, type -inf.

Enter an upper limit:

If you need oo, type inf.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

### Solution

Your input: find the area of the surface of revolution of $f\left(x\right)=x^{2}$ rotated about the x-axis on $\left[0,1\right]$

The surface area of the curve is given by $S = 2\pi \int_a^b f \left(x\right) \sqrt{\left(f'\left(x\right)\right)^2+1}d x$

First, find the derivative: $f '\left(x\right)=\left(x^{2}\right)'=2 x$ (steps can be seen here)

Finally, calculate the integral $S = \int_{0}^{1} 2 \pi x^{2} \sqrt{\left(2 x\right)^{2} + 1} d x=\int_{0}^{1} 2 \pi x^{2} \sqrt{4 x^{2} + 1} d x$

The calculations and the answer for the integral can be seen here.