Chapter 6 - Section 6.3 - Volumes by Cylindrical Shells - 6.3 Exercises - Page 466: 50

$\left( {\text{a}} \right)2\pi \int_0^3 {\left( {3{x^2} - {x^3}} \right)} dx,{\text{ }}\left( {\text{b}} \right)\frac{{27}}{2}\pi $

$$\eqalign{ & {\text{From the graph we have:}} \cr & y = 4x - {x^2}{\text{ and }}y = x \cr & {\text{Find the intersection points}} \cr & 4x - {x^2} = x \cr & {x^2} - 3x = 0 \cr & x\left( {x - 3} \right) = 0 \cr & x = 0,{\text{ }}x = 3 \cr & 4x - {x^2} \geqslant x{\text{ on the interval }}0 \leqslant x \leqslant 3 \cr & \cr & \left( {\text{a}} \right){\text{Using the shell method about the }}y{\text{ - axis}} \cr & V = \int_a^b {2\pi x\left[ {f\left( x \right) - g\left( x \right)} \right]} dx \cr & {\text{Let }}f\left( x \right) = 4x - {x^2}{\text{ and }}g\left( x \right) = x \cr & {\text{Therefore}}{\text{,}} \cr & V = \int_0^3 {2\pi x\left[ {4x - {x^2} - x} \right]} dx \cr & V = \int_0^3 {2\pi x\left( {3x - {x^2}} \right)} dx \cr & V = 2\pi \int_0^3 {\left( {3{x^2} - {x^3}} \right)} dx \cr & \cr & \left( {\text{b}} \right) \cr & {\text{Integrating}} \cr & V = 2\pi \left[ {{x^3} - \frac{1}{4}{x^4}} \right]_0^3 \cr & {\text{Evaluating}} \cr & V = 2\pi \left[ {{{\left( 3 \right)}^3} - \frac{1}{4}{{\left( 3 \right)}^4}} \right] - 2\pi \left[ {{{\left( 0 \right)}^3} - \frac{1}{4}{{\left( 0 \right)}^4}} \right] \cr & V = 2\pi \left( {27 - \frac{{81}}{4}} \right) - 0 \cr & V = \frac{{27}}{2}\pi \cr} $$