Why does my washer method setup give the wrong volume?

[IsabellaJ]

I was working through a problem about finding the volume of a solid of revolution using the washer method, and I got a different answer than the solution key. My integral setup seems correct, but I think my error might be in how I'm applying the fundamental theorem of calculus when evaluating the definite integral, specifically in handling the antiderivative of the squared terms after expansion.

[EdwardGT]

I ran into that with the washer method and the numbers looked off. I found the error hides in the antiderivative after expansion and keeping track of the limits. double check the sign when you subtract the two antiderivatives at the endpoints.

[BrandonCS]

I did a quick numeric check by plugging in small samples and the numbers still lined up with my setup and not the key which made me doubt the method rather than the algebra.

[JasonS]

Maybe the real problem is the limits or the radius function order leading to a sign slip in the outer minus inner part. It felt silly when I realized I had the terms inverted.

[Nora59]

If you want a concrete nudge try to write the antiderivative as a clean polynomial and plug in the endpoints one at a time using a calculator once to verify the arithmetic.