Calculating Volumes of Solids of Revolution

Imagine we have a sketch of a curve on the $xy$-plane. If we took this curve and rotated it in 3D around the $x$-axis, we can create the surface of a 3D object!

These 3D shapes are called solids of revolution, and they pop up everywhere from bottles and vases to car parts and architectural designs. If the shape is very simple, like a cylinder or a cone, we can use a formula to calculate the volume inside this objects, but if the shape is more complicated then we need a new tool.

It turns out that we can use integral calculus to calculate the volumes of these solids. For example, if we are performing a rotation around the $x$-axis, we can use the formula

$V = \int_a^b \pi f(x)^2 dx$

to calculate the volume of the solid between the planes $x=a$ and $x=b$

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