find the volume of the solid generated by revolving the region bounded by y=1/(1+x), y=0, x=0, & x=3 about y=4

Answer 1

The solid that is generated is like a donut - completely symmetrical around its axis. So, the easiest way to compute the volume is by integrating slices perpendicular to the axis.

The axis (y=4) is parallel to the x-axis, so the desired result is:

Integral from x=0 to x=3 of A(x)dx

Each slice is an annulus - an circle with a circle cut out of its middle - with center on the axis.

The area of a circle is (pi)r^2 so the area of an annulus is:

A(x) = (pi)(R(x)^2 - r(x)^2)

where:

R(x) is the outer radius of the annulus and r(x) is the inner radius.

The outer radius is the radius further from the axis. In this case, since y(x) is always greater than 0, the outer radius is determined by the line y = 0, and so is constant at 4.

The inner radius is the distance from the axis to y(x), or:
r(x) = 4 - 1/(1+x)

Now we have all the pieces, so we just have to put them together, compute the integral, and we have the answer.