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Since x goes from 0 to 1 in this region, the volume of the solid is
(b) Rotating R around the y-axis gives us the solid
A slice perpendicular to the y-axis is a washer with inner radius y and outer radius .
The area of the side of the washer is
and the volume of the washer is
π[y – y2] Δ y.
Since y goes from 0 to 1 in the region R, the volume of the solid is
Let R be the region bounded by y = x and y = x2. Find the volume of the solid obtained by rotating R around
(a) the line y = -1
(b) the line x = 4
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
router = x + 1.
Similarly, the inside of the solid is bounded by the line y = x2. 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
rinner = x2 + 1.
We have router = x + 1 and rinner = x2 + 1, so the volume of the washer is
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
router = 4 – x = 4 – y.
The inner boundary of the solid is the line y = x2, 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.