Let *R* be the region bounded by the graphs *y* = 1 – *x*, *y* = *x*^{2} + 1, and the line *x* = 1.

Find an integral expression for the volume of the solid obtained by rotating *R* around the line

(a) *y* = 0

(b) *y* = 2

(c) * y* = -2

(d) *y* = 5

Answer

The region *R* looks like this:

(a) When we rotate the region *R* around the line *y* = 0 we get the solid

We can see from the picture that the outer radius is the distance from the *x*-axis to the graph *y* = *x*^{2} + 1, and the inner radius is the distance from the *x*-axis to the graph *y* = 1 – *x*.

So

*r*_{outer} = *x*^{2} + 1

*r*_{inner} = 1 – *x*.

Since the thickness of a disk is *dx*, the volume of the disk is

π[(*x*^{2} + 1)^{2} – (1 – *x*)^{2} ] Δ *x*.

There are disks from *x* = 0 to *x* = 1. The volume of the solid is

(b) When we rotate *R* around the line *y* = 2 we get

The outer radius is the distance from the graph *y* = 1 – *x* to the axis of rotation *y* = 2.

This distance is

*r*_{outer} = 2 – (1 – *x*) = 1 + *x*

The inner radius is the distance from the graph *y* = *x*^{2} + 1 to the axis of rotation *y* = 2

This distance is

*r*_{inner} = 2 – (*x*^{2} + 1) = 1 – *x*^{2}

This means the volume of a disk is

π[(*r*_{outer})^{2} – (*r*_{inner})^{2}] Δ *x* = π[(1 + *x*)^{2} – (1 – *x*^{2})^{2} ] Δ *x*

Put the pieces together, we get the volume of the solid:

(c) When *R* is rotated around the line *y* = -2 we get the solid

The outer radius is the distance from the axis of rotation *y* = -2 to the graph *y* = *x*^{2} + 1.

This distance is

*r*_{outer} = (*x*^{2} + 1) + 2 = *x*^{2} + 3.

The inner radius is the distance from the axis of rotation *y* = -2 to the graph *y* = 1 – *x*.

This distance is

*r*_{inner} = (1 – *x*) + 2 = 3 – *x*.

The volume of a disk is

π[(*r*_{outer})^{2} – (*r*_{inner})^{2}] Δ *x* = π[(*x*^{2} + 3)^{2} – (3 – *x*)^{2} ] Δ *x*,

and the volume of the entire solid is

(d) Rotating *R* around the line *y* = 5 produces a solid with a very large hole in the middle.

Let's see if we can do this one without a picture.

The outer radius is the distance from the axis of rotation *y* = 5 to the graph *y* = 1 – *x*.

This is

*r*_{outer} = 5 – (1 – *x*) = 4 + *x*.

The inner radius is the distance from the axis of rotation *y* = 5 to the graph *y* = *x*^{2} + 1.

This is

*r*_{inner} = 5 – (*x*^{2} + 1) = 4 – *x*^{2}.

The volume of a disk is

π[(*r*_{outer})^{2} – (*r*_{inner})^{2}] Δ *x* = π[(4 + *x*)^{2} – (4 – *x*^{2})^{2} ] Δ *x*.

and the volume of the entire solid is