In order to practice doing exercises with integrals, I've tried quite a few exercises where I had to rotate a certain area around a certain axis in order to find the volume of a solid. Most of the time, bounds were either obvious, or given, however I've just stumbled upon a problem where I can't seem to get the right answer with the bounds I figured were right.
This is the problem: Find the volume of the solid generated by revolving the region bounded above by the graph of $y=2x-2x^2$ and below by the $x$−axis about the line $x = 2$.
The, let's call it, "weird" part of the problem is that I'm not asked to rotate the area given by the curve around the $x$-axis, but around $x=2$. However, $y$ always is smaller than $2$. Therefore, subtracting $2$ from the function (which gives $2x-2x^2-2$) results in a function that is negative and does not have any intersections with $x=0$. Since I could not really wrap my head around how the obtained solid may look like, I simply used the bounds $0$ and $1$ (zeros of the original function), but never got the right result.
The result should be $\pi$. I, however, got $15\pi/4$ with every single one of my tries: my integrals always looked similar to this one: $$ \pi\int_0^1 (2x-2x^2-2)^2 dx. $$
What am I doing wrong, and how should I try to "imagine" the obtained solid in order to maybe not commit the same errors should a similar problem arise?
Thank you very much.
There's a lot going on here. First, the region is a little hump between $0$ and $1$, above the $x$-axis and under the parabola. Second, I don't think you realize that the line $x=2$ is vertical. When you rotate the region about $x=2$, you get the top half of a donut. Such a volume is more suited for the method of shells than the method of washers.
The radius of each shell is $2-x$ and its height is $2x-x^2$, so the integral is
$$\int_0^1 2\pi(2-x)(2x-x^2) \; dx = \pi.$$
But I think your real problem is that you're just trying to apply a formula without understanding the formula. There are too many problems that don't fit the formulae given in the calc texts, so one needs to be able to adapt it to fit other situations. In fact, this is the main skill for that chapter in the calc text.