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.
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)`