Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

y=sqrt(x-1), y=0, x=7; about the x-axis

a) V=

b) Sketch the region.

c) Sketch the solid, and a typical disk or washer.

Answer 1

Given:

y = sqrt(x-1)

y = 0

x = 7 (don't confuse this with y=7, one of the answers is set up for that trap)

Based on the first equation we also know that

x>=1

To picture how this will look on a graph I would go through the following steps.

From the first equations:

x = y² + 1

The important part here is to recognize that this is similar to y = x² + 1. This means it will be of a parabolic shape, but opening to the right rather than upwards. The parabola will be bound by x=1 (left), x=7 (right), y=0(bottom), and x = y² + 1 (top). The region within these bounds is what will be integrated on the x-y axis.

To get the 3D picture, rotate the image above around the x-axis and create a vertical slice at any point.

Some items that we know.

1. Since one of the y boundaries given is the x=axis (y=0) and since we are rotating about the x-axis, the final 3D shape will be a solid without any voids.
2. Since the other y boundary is a parabolic equation, we know that the outer surface of the solid will be of that shape as well.