Volume of revolution
Spin a region around the x-axis. Slice it into thin discs of radius f(x), each with volume pi f(x)^2 dx, and add them up with an integral.
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Analogy
A potter's wheel: spinning a flat shape gives a solid vase.
Example
Radius is sqrt(x), so area = pi x.Volume = integral of pi x dx from 0 to 4.= pi [x^2/2] = pi (16/2).Volume = 8 pi, about 25.1.
In practice
Gives volumes of cones, spheres, bottles and tanks.
Practice
Flashcards
- What is the disc volume formula?V = pi times the integral of f(x)^2 dx.
Questions people ask
Why is the disc formula pi times f squared?
Each thin slice is a circle (disc) of radius f(x), and the area of a circle is pi times radius squared. Multiply by the slice thickness dx to get a volume. The integral adds up all slices from the start to the end.
What if the region has a hole in the middle?
Use washers: discs with a hole. The area of each slice is pi times outer radius squared minus pi times inner radius squared. Integrate that difference over the interval. This handles shapes such as a pipe or a ring. If the outer radius is 3 and the inner radius is 1, each slice has area 8 pi.