Why does shell method give a different constant than washers for this volume?

[Alexander.W]

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 washers. Both approaches seem valid for this shape, yet I’m getting a different constant in front of π.

[Aubrey.C]

If the region sits to the right of the y-axis, the shell setup uses radius x and height top minus bottom, with the x-range taken across the full region. If your result is off by a constant, you probably forgot to cover the full x-range or misapplied a symmetry factor and only integrated half the region.

[MatthewER]

With washers, you set up using outer radius R(y) and possibly inner radius r(y). If there’s a hole, you subtract. If the left edge isn’t on the axis, the inner radius isn’t zero, and forgetting that gives the wrong constant.

[OliviaXH]

I had a similar mismatch once; I redrew the region and rechecked the y-limits. The algebra was fine but the numbers wouldn’t line up until I tested a simple value to sanity-check the bounds.

[KevinMP]

Could the axis actually be the x axis in the problem instead of the y axis?