Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the -axis.

y = x^3,       y = 8,         x = 0

Answer 1

Step 1:

The curve equations are , and .

Cylindrical shells method:

The volume of the solid obtained by rotating about the -axis the region under the curve is

.

Find the point of intersections.

Substitute x = 0 in .

Now we integrate over the interval 0 to 8.

Step 2:

Let us assume the volume of the solid is obtained by rotating about -axis.

Consider .

Apply cube root on each side.

Volume of the solid obtained by rotating about the -axis the region under the curve and   is

Solution:

The volume of the solid is