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
$\endgroup$1 Answer
$\begingroup$You did a very strange job! $$\int (x^3-2x^2)^2 \, \mathrm{d}x\neq\int(x^5-2x^4) \, \mathrm{d}x$$ Indeed: $$\int (x^3-2x^2)^2 \, \mathrm{d}x=\frac{1}{7}x^7-\frac{2}{3}x^6+\frac{4}{5}x^5+C$$ so $$V=\int_0^2\pi y^2 \, \mathrm{d}x=\pi \cdot \left(\frac{1}{7}2^7-\frac{2}{3}2^6+\frac{4}{5}2^5\right)-\pi(0)=\pi\times\frac{128}{105}$$
