- Topics At a Glance
- Area
- Assumptions When Finding Area
- Triangles
- Circles
- Integrating with Polar Coordinates
- Volume
- Volumes of Solids with Known Cross-Sections
- Pyramids, Cones, and Spheres
**Volume of Solids of Revolution**- Disks and Washers
- Shells
- Washers vs. Shells
- Arc Length
- Arc Length for Parametric Functions
- Arc Length for Polar Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Solids of revolution are not fruits thrown at tyrannical rulers in protest. A **solids of revolution** is a volume built by rotating an area around a predetermined center line called the **axis of rotation**.

We mentioned before that way to think of this is like 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 football players practice plays before a big game, we are 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.

Answer.

As always, we will 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.

Example 1

Draw the solid obtained by rotating the region bounded by |

Exercise 1

Let *R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the *x*-axis

Exercise 2

Let *R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the *y*-axis

Exercise 3

Let *R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the line *x* = 1

Exercise 4

*R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the line *x* = 2

Exercise 5

*R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the line *x* = -1

Exercise 6

*R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the line *y* = 1

Exercise 7

*R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the line *y* = 4

Exercise 8

*R* be the region bounded by the graphs of , *x* = 1, and the *x*-axis. Draw the solid obtained by rotating *R* around

the line* y* = -1