Chapter 6 - Applications of Integration - 6.4 Volume by Shells - 6.4 Exercises - Page 443: 44

$$V = 500\pi $$

$$\eqalign{ & y = {\left( {x - 2} \right)^3} - 2,{\text{ }}x = 0,{\text{ }}y = 25{\text{ revolved about the }}y{\text{ - axis}} \cr & {\text{Using the washer method about the }}y{\text{ - axis}} \cr & V = \int_c^d {\pi \left[ {p{{\left( y \right)}^2} - q{{\left( y \right)}^2}} \right]} dy \cr & y = {\left( {x - 2} \right)^3} - 2 \to x = \root 3 \of {y + 2} + 2 \cr & {\text{From the graph of the region shown below}} \cr & x = \root 3 \of {y + 2} + 2 \geqslant 0{\text{ on the interval }}\left[ { - 10,25} \right]{\text{ }} \cr & {\text{Let }}p\left( y \right) = \root 3 \of {y + 2} + 2{\text{ and }}q\left( y \right) = 0 \cr & {\text{Therefore}}{\text{,}} \cr & V = \int_{ - 10}^{25} {\pi \left[ {{{\left( {\root 3 \of {y + 2} + 2} \right)}^2} - {{\left( 0 \right)}^2}} \right]} dy \cr & V = \pi \int_{ - 10}^{25} {\left[ {{{\left( {y + 2} \right)}^{2/3}} + 4{{\left( {y + 2} \right)}^{1/3}} + 4} \right]} dy \cr & {\text{Integrating}} \cr & V = \pi \left[ {\frac{{3{{\left( {y + 2} \right)}^{5/3}}}}{5} + 3{{\left( {y + 2} \right)}^{4/3}} + 4y} \right]_{ - 10}^{25} \cr & V = \pi \left[ {\frac{{3{{\left( {25 + 2} \right)}^{5/3}}}}{5} + 3{{\left( {25 + 2} \right)}^{4/3}} + 4\left( {25} \right)} \right] \cr & - \pi \left[ {\frac{{3{{\left( {25 - 10} \right)}^{5/3}}}}{5} + 3{{\left( { - 10 + 2} \right)}^{4/3}} + 4\left( { - 10} \right)} \right] \cr & {\text{Simplifying}} \cr & V = \pi \left( {\frac{{2444}}{5}} \right) - \pi \left( { - \frac{{56}}{5}} \right) \cr & V = 500\pi \cr & \cr & {\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{From the graph of the region shown below}} \cr & 25 \geqslant {\left( {x - 2} \right)^3} - 2{\text{ on the interval }}\left[ {0,5} \right]{\text{ }} \cr & {\text{Therefore}}{\text{,}} \cr & V = \int_0^5 {2\pi x\left[ {25 - {{\left( {x - 2} \right)}^3} + 2} \right]} dx \cr & V = 2\pi \int_0^5 {\left[ {25x - x{{\left( {x - 2} \right)}^3} + 2x} \right]} dx \cr & V = 2\pi \int_0^5 {\left( {35x - 12{x^2} + 6{x^3} - {x^4}} \right)} dx \cr & {\text{Integrating }} \cr & V = 2\pi \left[ {\frac{{35{x^2}}}{2} - 4{x^3} + \frac{{3{x^4}}}{2} - \frac{{{x^5}}}{5}} \right]_0^5 \cr & V = 2\pi \left[ {\frac{{35{{\left( 5 \right)}^2}}}{2} - 4{{\left( 5 \right)}^3} + \frac{{3{{\left( 5 \right)}^4}}}{2} - \frac{{{{\left( 5 \right)}^5}}}{5}} \right] - 2\pi \left( 0 \right) \cr & V = 2\pi \left( {250} \right) \cr & V = 500\pi \cr & {\text{Shell method is easier to apply}} \cr} $$