The question is as follows: Find the volume of the solid generated by revolving the region bounded by the parabola $y = x^2$ and the line $y = 1$ about the line $y = -1$.
My attempt: Using the washer method, I set the outer radius to $1 + x^2$ and the inner radius to be $1$. This gives me the integral:
$V = \pi\int_{-1}^1(1+x^2)^2-1^2dx = \pi\int_{-1}^12x^2+x^4dx$, which when evaluated gives the answer of $26\pi/15$. This however, is wrong. The correct answer is $64\pi/15$.
I put it through Wolfram Alpha and it gave me the same answer: https://www.wolframalpha.com/input/?i=rotate+the+region+between+0+and+x%5E2+with+-1%3Cx%3C1+around+the+line+y+%3D+-1
Any help would be appreciated.
EDIT: I tried using the shell method and got the correct answer, using $y+1$ as the radius and $\sqrt y$ as the height. So now the question is how would you do this using the washer method.
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