Solve Volume of Revolution Problem: y=x^{-2} to y=e Rotating Around Y-Axis

[vilhelm]

Problem
The area between $$y=x^{-2}$$ and x=1 & y=e is rotating around the y-axis. What is the volume?

Attempt
$$\pi\left( r_{outer\mbox{}} \right)^{2}\; -\; \pi \left( r_{inner\mbox{}} \right)^{2} \; \; \; \delta y$$.
$$\frac{1}{x^{2}}=y\; gives\; \frac{1}{y}=x^{2}\; and\; r=\sqrt{y}$$
$$V=\pi \int_{1}^{e}{\frac{1}{y}-1\; dy}$$

 

[clementc]

Can I assume that the question means the area above the x axis?

Anyway, firstly you must draw the graph. Are you trying to find the volume of the cylinder then subtract the volume not included in the region? If so isn't the x=1 the outer radius? And so then rather the inner radius should then be (1-x)

I find it easier and a nicer integral to just sum the volumes - so calculate the volume generated by the cylinder from y=0 to y=1, and then add to this the volume generated by the curve from y=1 to y=e. Hopefully this helps

 

[vilhelm]

Yes, that would be more effective. I am, however, trying to learn this formula, and would like to use it on this problem.

It is indeed the area over the x-axis. The upper bound is e, the right side is x=1, and the third line is x^-2. If you see what I mean…

I don't follow when you say "If so isn't the x=1 the outer radius? And so then rather the inner radius should then be (1-x)"

 

[clementc]

Sorry for lag reply. Oh my god I've just spent the last 30 min scribbling furiously away at my desk because I was getting different answers from 2 different methods and didn't know why. When I did your way I got it wrong because the algebra was more complicated. So YEAH I really recommend just doing it the simplest way.

I shall now guess what you're trying to do. Are you trying to find the volume of the cylinder from y=0 to e made by a rotation, and subtract from this the volume generated by the unwanted area (between y=1/x^2, y=e and x=1)?

Okay, if so, what you've done is switch the 1 and 1/y. When you draw the diagram (draw a nice big one!), you will see that x=1 is the OUTER radius, and the inner radius is x.
So an infinitesimal thickness disc would have volume $$dV = \pi (1^2-x^2)dy$$. Now since $$x=\frac{1}{\sqrt{y}}$$, then simply complete the integral, subtract this from the area of the cylinder, and the answer should be $$2\pi$$ cubic units!