Volumes of Solids of Revolution Method of Cylinders — Question 8

Question 8 – Solution

1. Identify the shell radius and height.

Use vertical slices over $0\le x\le1$, parallel to the $y$-axis:$$r(x)=x,\qquad h(x)=e^{-x}-0=e^{-x}.$$

2. Set up the volume integral.

Circumference times height times thickness gives$$V=2\pi\int_0^1r(x)h(x)\,dx=2\pi\int_0^1xe^{-x}\,dx.$$

3. Apply integration by parts.

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

4. Evaluate the definite integral.

Factor the antiderivative and substitute the bounds:$$\int_0^1xe^{-x}\,dx=\left[-(x+1)e^{-x}\right]_0^1=-\frac2e-(-1)=1-\frac2e.$$

5. Multiply by the circumference factor.

$$V=\boxed{2\pi\left(1-\frac2e\right)}.$$Since $e>2$, the expression is positive, as a volume must be.

All volumes are in cubic units.