How do you find the volume of the region bounded by #y=sqrt x,# and the lines #y=2# and# x=0# and it is revolved about the line #y=2#?
First, try and imagine how it would look. This could be described as a "cone" where the tracing of the surface starting from every point on the base's circumference (at the same time) grows like
Thankfully, this is a closed surface with a finite volume (not Gabriel's horn), due to the revolution occurring around
So the interval you would have to integrate over is
The volume is defined as:
Just pick one of them and use it, since we've redefined the revolution to occur using one of these symmetric curves around the x-axis. I picked the second one.
Comparing with a regular right cylindrical cone, we would expect:
So our answer is reasonable. You can also see here that our answer is identical to the result we would have gotten if we didn't shift the "cone":