Question

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. \(\begin{aligned}{}y = x,\;\\y = x{e^{1 - x/2}},\end{aligned}\)about \(y = 3\)

Short answer

The volume of the solid is \(V = 9.82476\).

Step-by-step solution

Volume of the washer

Volume is given by

\(V = \pi \int_0^2 {{\rm{ (Outer Radius}}{{\rm{)}}^2}} - {({\rm{ Inner Radius }})^2}\;{\rm{d}}x\)

Blue curve: \(y = x\)

Green curve: \(y = x{e^{1 - x/2}}\)

The volume of the solid

Rotate a thin vertical strip as shown in the figure, about the line \(y = 3\) to get a washer with inner radius \(\left( {3 - x{e^{1 - x/2}}} \right)\) and outer radius \((3 - x)\)

The volume of the washer is given by the following formula:

\(V = \pi \int_0^2 {{\rm{ (Outer Radius}}{{\rm{)}}^2}} - {({\rm{ Inner Radius }})^2}\;{\rm{d}}x\)

After necessary substitutions we get:

\(\begin{aligned}{}V = \pi \int_0^2 {{{(3 - x)}^2}} - {\left( {3 - x{e^{1 - x/2}}} \right)^2}\;{\rm{d}}x\\ = 24e - \frac{{142}}{3} - 2{e^2}\\V \approx 9.82476\end{aligned}\)

Since the problem is marked CAS, use a calculator to find the integral.