Volume of a Solid Integral Question

Question

Consider the (unbounded) region R in the first quadrant consisting of the area under the curve y=1 and to the left of the curve x=ln(e/y)

Let "V" be the volume of the solid generated by revolving R about the y - axis. Set up two expressions for V, one that uses the slicing/washer method and one that uses the cylindrical shells method. Choose one of these methods to determine the exact value of V.

Answer 1

First draw the region with a graphing calculator, for example Desmos.com

1.

Slicing into disks parallel to the x axis of thickness dy and radius x(y):

Volume = integral πx(y)² dy for y = 0 to 1. Plug in the expression for x(y) = ln(e/y) = 1 - lny. The integral will probably require integration by parts, so I don't recommend attempting it.

2.

Slicing into cylindrical shells of radius x, thickness dx, height y(x):

Volume = integral 2πx y(x) dx for x = 0 to infinity.

This integral has to be split into two integrals because y(x) has different expressions over different x intervals.

For x = 0 to 1, y(x) = 1 (see your graphing calculator plot)

For x = 1 to infinity, solve the equation x = ln(e/y) = 1 - lny for y: y(x) = e^1-x

These two integrals seem easier to evaluate than the integral in part 1.