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Volume of Solids of Revolution

Solids of revolution aren't fruits thrown at tyrannical rulers in protest. A solid of revolution is a 3D object built by rotating an area around a predetermined center line called the axis of rotation.

We mentioned before that one way to think of this is as a bundt cake. If we aren't happy with a thin slice, we can choose a thicker piece by making an initial slice and rotating the knife around the center to cut out a better sized portion. If we were to rotate all the way around once without cutting, we would form the entire cake, which is a full revolution of the solid cake.

Understanding how to make solids of revolution can be difficult to picture, so drawing them is often very useful. Just like how football players practice plays before a big game, we're going to get plenty of practice drawing solids of revolution before we build integrals with them.

Sample Problem

Draw the solid obtained by rotating the region bounded by y = x, y = 1, and the y-axis around the y-axis.