We're getting pretty familiar with this region by now: (a) When we rotate *R* around the line *y* = -1, we get a solid with a hole in its middle. This means a slice of this solid will look like a washer. The outside of the solid is bounded by the line *y* = *x*. However, since the center axis of the solid is *not* the *x*-axis, the outer radius is not just *x*. If we look at the picture we can see that the outer radius is *r*_{outer} = *x* + 1.
Similarly, the inside of the solid is bounded by the line *y* = *x*^{2}. Since the center axis of the solid is at *y* = -1, which is a distance of 1 away from the *x*-axis, we have to add 1 to get the inner radius *r*_{inner} = *x*^{2} + 1.
We have *r*_{outer} = *x* + 1 and *r*_{inner} = *x*^{2} + 1, so the volume of the washer is π[(*r*_{outer})^{2} – (*r*_{inner})^{2}] Δ *x* = π [(*x* + 1)^{2} – (*x*^{2} + 1)^{2} ] Δ *x*. The variable *x* goes from 0 to 1 in the region, so the volume of the solid is (b) When we rotate *R* around the line *x* = 4 we get the solid Slices perpendicular to the line *x* = 4 are washers. The outer boundary of the solid is at the line *y* = *x*. However, the center line of the solid is at *x* = 4. By looking at the picture we can see that the outer radius of the slice at height *y* is *r*_{outer} = 4 – *x* = 4 – *y*.
The inner boundary of the solid is the line *y* = *x*^{2}, or . By looking at the picture, we can see that the inner radius of the slice at height *y* is . The volume of the slice is The variable *y* goes from 0 to 1 in the original region *R*, so the volume of the solid is We're being bombarded with a multitude of pitched frisbee disks because it's important. Even the most experienced volume-making, cake baking, frisbee tossing expert draws pictures to avoid making mistakes. |