I'm having trouble trying to see what the region I'm supposed to be computing looks like.
The volume of the solid obtained by rotating the region enclosed by
$$y = 1/x^5 , y=0, x=3, x=4$$
about the line $x=−4$ can be computed using the method of cylindrical shells via an integral. What does the integrand look like?
Given all the parameters I decided to visualize this first with the graphs. Here are all the curves:
So everything is being rotated around the black line on the left, $x=-4$. But there are many things that just don't make sense to me:
- The red function, $y=1/x^5$ isn't even well defined at $x=0$, as it approaches infinity. How is this region "enclosed" at all?
- The purple line on the far right, $x=4$, doesn't seem to contribute anything to the enclosure, does it? It's just a step after $x=3$ which already limits the volume of the solid.
- Whatever this solid is, it doesn't seem to be enclosed in any way whatsoever to me. I see that it is limited by $y=0$ at the bottom, but there's nothing bounding it at the top as all the curves just continue straight up.
I already know that the answer is
$$\int\frac{2\pi(4+x)}{x^5}dx$$
But I can't see the reasoning behind the question at all. What does the region look like, and how is it enclosed?
You might find Calcplot3D useful for this type of problem. You can add a surface of revolution, and with enough manipulation of the axes/scales you can produce plots like this:
It will also be useful to have the rotation and key shortcuts page for reference.