Chapter 7 - Applications of Integration - 7.2 Exercises - Page 454: 29

$$V = \frac{{277}}{3}\pi $$

$$\eqalign{ & y = {x^2} + 1,{\text{ }}y = - {x^2} + 2x + 5,{\text{ }}x = 0,{\text{ }}x = 3 \cr & {\text{Let }}y = y \cr & {x^2} + 1 = - {x^2} + 2x + 5 \cr & {\text{Rearrange}} \cr & 2{x^2} - 2x - 4 = 0 \cr & {x^2} - x - 2 = 0 \cr & {\text{Factoring}} \cr & \left( {x - 2} \right)\left( {x - 1} \right) = 0 \cr & x = 1,{\text{ }}x = 2 \cr & {\text{From the graph }} - {x^2} + 2x + 5 \geqslant {x^2} + 1{\text{ on }}\left[ {0,2} \right] \cr & {\text{From the graph }}{x^2} + 1 \geqslant - {x^2} + 2x + 5{\text{ on }}\left[ {2,3} \right] \cr & {\text{Apply the washer method}} \cr & V = \pi \int_a^b {\left( {{{\left[ {R\left( x \right)} \right]}^2} - {{\left[ {r\left( x \right)} \right]}^2}} \right)} dx,{\text{ then}} \cr & V = \pi \int_0^2 {\left( {{{\left[ { - {x^2} + 2x + 5} \right]}^2} - {{\left[ {{x^2} + 1} \right]}^2}} \right)} dx \cr & + \pi \int_2^3 {\left( {{{\left[ {{x^2} + 1} \right]}^2} - {{\left[ { - {x^2} + 2x + 5} \right]}^2}} \right)} dx \cr & {\text{Simplifying}} \cr & V = \pi \int_0^2 {\left( {{x^4} - 4{x^3} - 6{x^2} + 20x + 25 - {x^4} - 2{x^2} - 1} \right)} dx \cr & \pi \int_0^2 {\left( { - {x^4} - 2{x^2} - 1 - {x^4} + 4{x^3} + 6{x^2} - 20x - 25} \right)} dx \cr & V = \pi \int_0^2 {\left( { - 4{x^3} - 8{x^2} + 20x + 24} \right)} dx \cr & + \pi \int_2^3 {\left( {4{x^3} + 8{x^2} - 20x - 24} \right)} dx \cr & {\text{Integrating}} \cr & V = \left[ { - {x^4} - \frac{8}{3}{x^3} + 10{x^2} + 24x} \right]_0^2 \cr & + \pi \left[ {{x^4} + \frac{8}{3}{x^3} - 10{x^2} - 24x} \right]_2^3 \cr & V = \left[ { - {{\left( 2 \right)}^4} - \frac{8}{3}{{\left( 2 \right)}^3} + 10{{\left( 2 \right)}^2} + 24\left( 2 \right)} \right] - 0 \cr & + \pi \left[ {{{\left( 3 \right)}^4} + \frac{8}{3}{{\left( 3 \right)}^3} - 10{{\left( 3 \right)}^2} - 24\left( 3 \right)} \right] \cr & - \pi \left[ {{{\left( 2 \right)}^4} + \frac{8}{3}{{\left( 2 \right)}^3} - 10{{\left( 2 \right)}^2} - 24\left( 2 \right)} \right] \cr & {\text{Simplifying}} \cr & V = \frac{{152}}{3}\pi - 0 - 9\pi + \frac{{152}}{3}\pi \cr & V = \frac{{277}}{3}\pi \cr} $$