Imagine cutting the object that you'd like to find the volume of straight down the middle, making a cross section. Look at one half of this cross section. For the technique to work, the edge of this half needs to be a straight line or a curve. For example, if you cut a sphere down the middle, one half of the cross section would be half a circle or a semicircle. If you cut a cone down the middle, one half of its cross section would be a straight line.
Choose the equation of the line or curve that you found in Step 1. To illustrate, use a basic straight line of y = x. This is the cross section of a cone that has an angle of 90 degrees at its tip.
Find the limits of the integral. Generally, these will be zero and the height of the shape. If the cone from Step 2 has a height of 5, then the limits of the integral would be 0 and 5.
Square the equation. For the cone, x would become x^2.
Take the integral of the squared equation and multiply it by pi. This changes the integration from being the sum of the areas of many thin rectangles to it being the sum of the volumes of many thin discs in which the function is the radius of the disc and dx is its width. The example is now pi times the integral of x^2 from 0 to 5.
Evaluate the integral. The antiderivative of x^2 is (1/3)x^3, so the integral is pi times (1/3)5^3 minus 0, which simplifies to (125/3) times pi, or approximately 130.9.