Question:
Find the volume of the solid formed by rotating completely about $x$-axis the area enclosed by a curve.
My answer:
I drew the curve and the area formed it is between $-2$ and $0$ (the rest of the curve after zero goes to infinite)
I calculated:
$$V=\pi\int_{-2}^0[x^3-2x^2]^2\,\mathrm dx=\pi\int_{-2}^0x^5-2x^4=\pi\int\dfrac{x^6}6-\dfrac{2x^5}5$$
I plug in the values (as shown bellow) and I get $-23\pi$ the correct answer should be $128\pi/105$. What am I doing wrong here? thank you
UPDATE: the region bound is actually between 2 and 0 not -2 and 0