Why did my shell method volume differ from washers for rotation about y-axis?

[Charles57]

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’m not sure why my approach gave a different result, and I’m wondering if I misidentified the radius or the height of a cylindrical shell.

[Paisley_T]

Shells around the y axis can fool you if the bounds aren’t lined up with x correctly. The distance from the axis is the x value, not a y. The height of each shell is the vertical length of the region at that x, so top y minus bottom y. If your top and bottom curves got swapped, you’ll get a different number than the washer setup.

[Emma92]

I’ve had the same mismatch. I compared with a washer setup and realized I mixed up which curve was on top for some x-intervals, so the height was off. When that happened, the whole integral looked right but the numbers didn’t match the key. Rechecking which function is larger for each x fixed it for me sometimes.

[Mia_L]

It may also be that the region isn’t simply two curves but includes a missing chunk so the washer version needs an inner radius. I remember drawing a quick sketch and realizing the hole you’d get with a washer didn’t exist with shells. That kind of mismatch often trips people up more than algebra.

[EthanFP]

Are you sure the axis is the y axis and not the line you were given in the problem?