Missing pictures and wrong solution?
Never mind, I see the method you are referring to in the textbook (even though I think what I wrote above is easier), I see that the students probably learn this cylindrical shell method. Still, I was wondering why the answers were different. If you use your thin shell method then you still have to consider the volume of the cylinder of revolution above y=-2 and below the x-axis. This cylinder has radius 2 and height 2. Therefore the volume is . Therefore the total volume is like the answer I got above.
I disagree, the question specifies the region bounded by the x-axis, so that central cylinder should not be included.
Oh I see how it's supposed to go now, it's like an annulus region, I interpreted it as to cut off the parabola at the x-axis and then take that whole shape and rotate it, makes sense now. The pictures still need to be fixed though.
Never mind, working now!
It might be easier to use disks ("washers" in this case, since there's a hole in the middle. I'm going to work on a solution doing that.
That's the point I was making with my first post, that disks are the more intuitive result and the integral is a million times easier. The integral is just the one that appears in my first post (minus the by taking out the part below the x-axis.
I flagged the new solution as good. I know some students get hung up on some of these geometrical (shell vs. disks vs. washers) ideas so if there were to be any changes, I'd recommend an extra graphic like in solution 1 that kind of shows a sample washer. It's not a big deal though.
Unfortunately I don't know how David made the pretty pictures. Is it worth swapping the two solutions, since the washers one is so much easier?
My instincts say yes but it might be a question best left for a 101 instructor, perhaps they drive the shell method into their brains like crazy and so that will be the first choice for students regardless of the difficulty. However, if they cover both methods equally then I would definitely vote this one first, power rule integrals always beat integration by parts in my books.