Find the volume of the solid formed by rotating the region enclosed by   y=e^(5x) , \ y=0, \ x=0, \ x=0.8 about the x-axis. Use integrals to find the volume.

Expert Answers
mlehuzzah eNotes educator| Certified Educator

Rotate the region around the x axis.  Imagine vertical "slices" cutting the shape into thin, disk-like slices.  We want to find the volume of each slice, and then add up all of the slices.  If we imagine thinner and thinner slices, (taking the limit as `n->oo` ) we get a Riemann sum:

Look at a slice located at x.  The radius of that slice is `e^(5x)` .  The thickness of that slice is `dx` .  Using the formula for the volume of a cylinder, we have that the volume of the slice is `pi (e^(5x))^2 dx`   , or `pi e^(10x)dx`

We want all of the slices from x=0 to x=.8

So we get:

`int_0^(.8) pi e^(10x) dx`

`=pi (1/10) e^(10x) |_0^(.8)`

`=(pi/10) (e^8 - e^0)`