Volumes of Solids of Revolution Method of Cylinders — Question 5

Question 5 – Solution

1. Identify the shell dimensions.

Use vertical slices for rotation about the $y$-axis. On $0\le x\le2$,$$r(x)=x,\qquad h(x)=xe^{-x}.$$

2. Set up the volume integral.

$$V=2\pi\int_0^2 r(x)h(x)\,dx=2\pi\int_0^2 x^2e^{-x}\,dx.$$

3. Integrate by parts once.

Let $u=x^2$, $dv=e^{-x}dx$, so $du=2x\,dx$ and $v=-e^{-x}$:$$\int x^2e^{-x}\,dx=-x^2e^{-x}+2\int xe^{-x}\,dx.$$

4. Integrate by parts a second time.

For the remaining integral, choose $u=x$ and $dv=e^{-x}dx$:$$\int xe^{-x}\,dx=-xe^{-x}+\int e^{-x}\,dx=-xe^{-x}-e^{-x}.$$Substitution gives $F(x)=-e^{-x}(x^2+2x+2)$.

5. Evaluate at $2$ and $0$.

$$\begin{align*} V&=2\pi\left[-e^{-x}(x^2+2x+2)\right]_0^2\\&=2\pi\bigl[-10e^{-2}-(-2)\bigr]\\&=2\pi(2-10e^{-2})=\boxed{4\pi(1-5e^{-2})}. \end{align*}$$

All volumes are in cubic units.