Wait, disk method? I'm forgetting about the shell method, aren't I?
So I'm asked to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis (obv.). I'm given:
x + 3 = 4y - y^2 , x = 0; about the x-axis
The book says the answer is (16π)/3
This seems weird to me because I'm imagining the graph will yield a solid just begging me to apply the washer method and use dx. Yet, unless my algebra is terrible (which wouldn't surprise me), I don't see how we can set an equation equal to one y value. Therefore, I tried devising a few approaches using dy and the disk method (I haven't seen a prototype for this problem, but I figured it couldn't be too hard to come up with my own approach to a textbook problem).
To start, I set x = -y^2 + 4y - 3. Then, I proceeded with π∫ (-y^2 + y - 3)^2 dy from 1 to 3 (value obtained by factoring). I ended up with (16/15)π. Way wrong.
I saw a problem that sort-of, kind-of resembled this one earlier in the book, which advised I solve the problem with the formula 2π∫ y[ u(y) - v(y) ]dy.
There wasn't an explanation though, so I figured maybe that y outside the brackets represented an equation composed with y variables. So I basically repeated my first attempt with 2π out front. Not surprisingly, wrong again.
The closest I came was trying 2π∫ (-y^2 + 4y - 3) dy (incorrectly assuming that maybe the 2 coefficient would account for the volume and I could discard the square outside the parentheses around the quadratic). I got 8π/3.
Then I figured that first y in 2π∫ y[ u(y) - v(y) ]dy could be replaced by the value obtained after applying the quadratic formula to -y^2 + 4y - 3. This led to me repeating that last equation with a 4 in front of the parentheses. I got 32π/3.
I guess I'd get 16π/3 if I'd have put a 2 in front of the parentheses, yet I can't figure out any good reason why I should.
I thought I was pretty decent at rotational solids, but this shook my confidence. I spent all afternoon thinking about this and I'm stumped. I apologize for the length of the post, but I figured you might appreciate the effort. Hopefully I presented everything clear enough and I hope someone can help!
Thanks a million in advance.