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.
Draw the solid obtained by rotating the region bounded by y = x, y = 1, and the y-axis around the y-axis.
As always, we'll start by drawing the area we are going to make the volume with:
We can label the axis of rotation and draw a mirror copy of the region on the other side of the axis of rotation. Drawing the mirror copy helps us picture the axis of rotation more clearly. We know the y-axis is the axis of rotation because the problem said to rotate around the y-axis.
Next, we draw curved lines, solid in front and dotted in back, to make it look like the region went all the way around the y-axis:
This will usually be good enough to give you an idea of what the solid looks like. In this case we ended up with a cone, but different regions and different axes of rotation will produce radically different solids.