Help with a calculus question?

What have you tried so far? If you let us know what you have tried and where you get stuck, we can help you. We aren’t supposed to simply do a problem for you (and you wouldn’t learn as much if we did).

If you’re using a calculus textbook, you should find an example that resembles the problem. If you need one, you can use an online resource like Paul Dawkin’s notes.

Yes, to second @bobbinhood, we cannot simply tell you how to do a homework problem on this site. We can help lead you there, but to do that we need some idea of what you already know. Do you know how to begin this problem? How do you think you should start?

Here at Fluther they’re very touchy about homework help. If I worked the whole problem out for you they’d have my tentacles! I think I can tell you, however, that the volume of a solid figures of revolution can be considered by adding the areas of infinitesimally thin disks stacked upon one another, each of whose radius can be calculated. In other words, integrate circular area along the axis of rotation. You should already know the shape of the defined region in the x-y plane prior to rotation. Hope that’s not too cryptic for you or over-the-line for me.

To do this as a calculus problem, you can use the shell method of finding volume. The formula for the shell method is to integrate 2 pi xf(x) dx. In the first problem, f(x) would be 2 – 2x, since that is the length of the piece being rotated about the y axis.

@LostInParadise The shell method works too. You can either integrate the areas of circular disks stacked longitudinally, or the areas of cylindrical shells nested along a radius. They are both useful approaches. Volumes or surfaces of revolution are 3-dimensional, but perfect circular symmetry reduces them to functions of 2 variables. So you integrate some kind of 2-dimensional region with respect to a 3rd orthogonal axis. Properly expressing boundaries is usually the tricky part.

Regions bounded by straight lines before rotation give rise to some combination of cylinders & cones, as @prasad points out, so you could just use solid geometry volume formulas. Functions in general, however, require the power of calculus to calculate volumes & areas, as illustrated by @LostInParadise‘s setup above.

In my day the book was Kreyszig Advanced Engineering Mathematics, apparently still in print today.