Volume of revolution between lines

If we have two lines, y = f(x) and y = g(x), where f(x) \geq g(x) for all x in the interval we’re interested in, and we want to find the volume of revolution when we revolve the area between those lines around the x-axis, we can either:

• find the volume of revolution of the area under y = f(x) and subtract the volume of revolution of the area under y = g(x), or
• find \pi \int_a^b (f(x)^2 - g(x)^2) \, dx

Formula

As just mentioned, we have a fancy formula for finding the volume of revolution of the area between two lines or curves, which is:

V = \pi \int_a^b (f(x)^2 - g(x)^2) \, dx

Where:

• V is the volume of revolution of the area between the two lines or curves.
• f(x) is the function that represents the upper line or curve (the one that’s further from the x-axis at the region we care about).
• g(x) is the function that represents the lower line or curve (the one that’s closer to the x-axis at the region we care about).
• a and b are the limits of integration, which represent the range of coordinates we’re rotating.

Common mistakes

• You cannot square the whole integral. You need to square the functions first, and then integrate the difference of the squares.
• You cannot just integrate the difference of the functions, and then square the result. You need to square the functions first, and then integrate the difference of the squares.
• Forgetting to multiply by \pi at the end!
• Forgetting to subtract the smaller function from the larger function. You need to make sure that f(x) \geq g(x) for all x in the interval you’re interested in, otherwise you’ll get a negative volume, which doesn’t make sense!