Rotating Curves around the X-Axis

Brody Eller

May 22, 2018

You can use the integral of a curve to calculate the volume of an area by rotating it around an axis.

When you find an integral, you find the area under a curve. If you were to take an integral at a single point on a curve, you’d get it’s distance from the x-axis. You can then take this distance and put it into the equation for area of a circle, .

Area of a Circle Revolution

Building on this, if you find a definite integral across a certain range, you can use that value to find the volume of a solid revolved around the x-axis.

Area of a Circle Revolution

To use a definite integral in the equation for area of a circle, replace r with the definite integral resulting in this:

where x represents the function of the curve. Then, solve the expression for the volume of the solid formed by rotating the curve around the x-axis.

Examples

  1. Find the volume of the solid formed by rotating the equation around the x-axis from to .
    1. Plug into the formula for area of a circle:
    2. Evaluate the definite integral:
  2. Find the volume of the solid formed by rotating the equation between and .
    1. Plug into the formula for area of a circle:
    2. Evaluate the definite integral: