Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 2, and x = 1 is revolved around the line y = -2.

Question

Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 2, and x = 1 is revolved around the line y = -2.

Answer 1

First, you would need the center of area y-bar of the region bounded by the function y = ln(x), x = 1 & y = 2

This is ∫∫ y dy dx / ∫∫ dy dx where the limits of integration for both integrals is 1< x <e² (e² if where the 2 curves meet or when y=2 and y=ln(x) intersect) and the y limits of integration are ln(x)< y <2.

You will need to know the bottom integral as this the area (A) between the curves.

Once you have y-bar, find the Distance (R) from y-bar to -2.

Then the volume of revolution is V = 2pi(R)(A)