$y= x^2$, $y= 6x-2x^2$
rotated about the $y$-axis. and this is in the cylindrical shells chapter of my book.
so I solve $x^2 = 6x-2x^2$ and I get $x = 0$ and $x= 2$
so volume for cylindrical shells is = to the integral of (circumference)(height)(thickness), correct?
so I have $2\pi$ integral of $\int x(6x-3x^2) dx$
I got $6x-3x^2$ by subtracting inner function from outer
so I went ahead and integrated $\int6x^2 - 3x^3 dx$ and got
$3x^3 - \frac{3}{4}x^4$ from $2$ to $0$
which gave me an answer of $12$, which I multiplied by $2\pi$ to get $24\pi$.
But my book is telling me the answer is $8\pi$. How did they get this?