Area, Volume, and Arc Length Exercises

Answer

When we rotate R around the line x = 2 we get the solid

The shell at position x has height . Its radius is the distance from the curve  to the axis of rotation x = 2.

The radius is

2 – x.

The volume of the shell is

The shells go from x = 0 to x = 1. This means we need to integrate from 0 to 1 to get the volume of the solid, or

Answer

When we rotate R around the line y = 1 we get the solid

The shell at height y looks like this. The radius of the shell is the distance from the edge of the cylinder (at y) to the axis of rotation (y = 1). So

radius = 1 – y.

The height of the shell is the distance from the curve  to the line x = 1, or

height = 1 – x = 1 – y².

The thickness of the shell is Δ y, so the volume of the shell is

2π(1 – y)(1 – y²) Δ y.

Since the shells go from y = 0 at the outside of the solid to y = 1 at the inside of the solid, the volume of the solid is

That's enough of that region. Let's do some problems with a different one, just to mix things up.

Answer

The region R looks like

When we rotate the region R around the y-axis we get the solid

Since the y-axis is the axis of rotation, the shells will have their opening around the y-axis. The shell at position x looks like

The radius of the shell is the distance from the edge of the shell to the y-axis, so

radius = x.

The height of the shell is the distance from the curve y = x² to the line y = 4, or

height = 4 – x².

The volume of a shell is

2π(x)(4 – x²) Δ x.

The variable x goes from 0 at the center of the solid to 2 at the outside of the solid, so the volume of the solid is

Answer

The region R looks like

When the region R is rotated around the line x = -1, we get a solid that looks like

The shells run parallel to the axis of rotation, so a shell at position x will look like

The radius of the shell is the distance from the edge of the shell to the axis of rotation x = -1, so

radius = x + 1.

The height of the shell is the distance from the curve y = x² to the line y = 4, which is

height = 4 – x²

(the height is the same as in the previous problem). The volume of a shell is

2π(x + 1)(4 – x²) Δ x.

Shells go from x = 0 at the inside edge of the solid to x = 2 at the outside edge of the solid, so the volume of the solid is

Answer

The region R looks like

When the region R is rotated around the x-axis we get

The shell at position y is centered around the axis of rotation:

The radius of the shell is y, and the height of the shell is . The volume of the shell is

The shells go from y = 0 at the inside of the solid to y = 4 at the outside of the solid, so the volume of the solid is

Answer

The region R looks like

When the region R is rotated around the line y = 5, we get the solid

The shell at position y looks like

The radius of the shell is the distance from the edge of the shell to the axis of rotation y = 5, so

radius = 5 – y.

The height of the shell is x, which is the same as . So the volume of the shell is

The shells still go from y = 0 to y = 4, so the volume of the solid is