Find the volume of the solid obtained by the revolving the region bounded by the graphs of the functions y = e^x, x = 2, x = 8, y = 0 about the x-axis. Draw a picture to illustrate the computation.

Question

Show all work.

Answer 1

So for your picture, you'll want to graph the two function for 2<x<8; this is simply an exponential function with y growing to about 3000 towards the end. Then for the rotation, you'll take a vertical rectangular subsection and rotate that around the x axis. The volume of this subsection will be the area of the circle (A = πr² = πe^2x) times the width of the rectangle (Δx or dx). So the total volume is the sum of all these rectangles: this is the definition of integration. Yefim has the solution to this integration in his response.

Answer 2

By disk method: v = π∫₂⁸e^2xdx = πe^2x/2₂⁸ = π(e¹⁶ - e⁴)/2 = 13958184