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I’m working through a calculus problem where I need to find the volume of a solid formed by rotating a region around the y-axis. I set up my integral using the shell method, but when I check the answer key, they used the washer method instead. I can’t seem to get my shell method setup to match their final result, and I’m wondering if I’m misidentifying the radius or height of a cylindrical shell.

Shell method around the y axis means the radius is the distance to the y axis. So r = x. The height is the vertical extent of the region at that x, so h(x) is top y minus bottom y. If you get a mismatch, recheck which curve sits on top for each x and the x-limits that sweep the whole region. A lot of times the mistake is mixing up height with a horizontal distance or forgetting that the region may be bounded on the left by x = a and on the right by x = b. Make sure your a and b cover the entire region.

I tried this once and kept getting a different number than the key. The problem hinges on the height h(x): if the region is between y = f(x) and y = g(x) you want h(x) = f(x) - g(x) in the right order, and you must keep the right signs when the curves cross. Also check the orientation of the region: if the region touches the y axis you might need to split the integral.

Sometimes the washer form is cleaner because you end up with R(y) and r(y) that come straight from solving x as a function of y. If the region is bounded by x = something and x = something else, you can get the outer and inner radii directly in terms of y. So the mismatch might just be you trying to force a shell setup when the problem actually wants washers.

Could the region actually be best described with horizontal slices? If you rotate around the y axis those give washers with radii depending on y. It might be that your shell radius x is correct but your height was wrong or your bounds. Do you know the exact region boundaries?

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