Rotating Curves around the X-Axis

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, $\pi r^2$.

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.

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 $f(x)=x^2$ around the x-axis from $x=0$ to $x=4$.
1. Plug $x^2$ into the formula for area of a circle: $\pi \int^{4}_{0}{(x^2)^2}$
2. Evaluate the definite integral: $\pi \frac{x^5}{5} \Bigr\|_{0}^{4} \approx 643.3982$
2. Find the volume of the solid formed by rotating the equation $f(x)=\sin{x}$ between $x=0$ and $x=\pi$.
1. Plug $\sin{x}$ into the formula for area of a circle: $\pi \int_{0}^{\pi}{\sin^2{x}}$
2. Evaluate the definite integral: $\frac{\pi\Bigl(x-\frac{\sin(2x)}{2}\Bigl)}{2} \Biggr\|_{0}^{\pi} \approx 4.9348$