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Volume of revolution between lines

If we have two lines, and , where for all 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 -axis, we can either:

  • find the volume of revolution of the area under and subtract the volume of revolution of the area under , or
  • find

Formula

Section titled “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:

Where:

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

Common mistakes

Section titled “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 at the end!
  • Forgetting to subtract the smaller function from the larger function. You need to make sure that for all in the interval you’re interested in, otherwise you’ll get a negative volume, which doesn’t make sense!