Chapter 7 - Integration Techniques - 7.1 Basic Approaches - 7.1 Exercises - Page 515: 62

$$\eqalign{ & \left( {\text{a}} \right)V = \frac{3}{{10}}\pi \cr & \left( {\text{b}} \right)V = \left( {4\ln \left( {\frac{2}{5}} \right) + \frac{{21}}{5}} \right)\pi \cr} $$

$$\eqalign{ & f\left( x \right) = \frac{1}{{x + 2}}{\text{ and the }}x{\text{ - axis on the interval }}\left[ {0,3} \right] \cr & \cr & \left( {\text{a}} \right){\text{Calculate the volume using the disk method about the x - axis}} \cr & V = \int_a^b {\pi f{{\left( x \right)}^2}dx} \cr & V = \pi \int_0^3 {{{\left( {\frac{1}{{x + 2}}} \right)}^2}dx} \cr & V = \pi \int_0^3 {\frac{1}{{{{\left( {x + 2} \right)}^2}}}dx} \cr & {\text{Integrating}} \cr & V = \pi \left[ { - \frac{1}{{x + 2}}} \right]_0^3 \cr & V = \pi \left[ { - \frac{1}{{3 + 2}} + \frac{1}{{0 + 2}}} \right] \cr & V = \pi \left( {\frac{3}{{10}}} \right) \cr & V = \frac{3}{{10}}\pi \cr & \cr & \left( {\text{b}} \right) \cr & f\left( x \right) = \frac{1}{{x + 2}} \cr & x = 0 \Rightarrow y = \frac{1}{2} \cr & x = 3 \Rightarrow y = \frac{1}{5} \cr & y = \frac{1}{{x + 2}} \cr & {\text{Solve for }}x \cr & xy + 2y = 1 \cr & xy = 1 - 2y \cr & x = \frac{1}{y} - 2 \cr & f\left( y \right) = \sqrt {{y^2} - 1} \cr & {\text{using the disk method about the y - axis}} \cr & V = \int_c^d {\pi f{{\left( y \right)}^2}dx} \cr & V = \int_{1/5}^{1/2} {\pi {{\left( {\frac{1}{y} - 2} \right)}^2}dy} \cr & V = \pi \int_{1/5}^{1/2} {\left( {\frac{1}{{{y^2}}} - 4\left( {\frac{1}{y}} \right) + 4} \right)dy} \cr & V = \pi \int_{1/2}^{1/5} {\left( {\frac{1}{{{y^2}}} - \frac{4}{y} + 4} \right)dy} \cr & {\text{Integrating}} \cr & V = \pi \left[ { - \frac{1}{y} - 4\ln y + 4y} \right]_{1/5}^{1/2} \cr & V = \pi \left[ { - \frac{1}{{1/2}} - 4\ln \left( {\frac{1}{2}} \right) + 2} \right] - \pi \left[ { - \frac{1}{{1/5}} - 4\ln \left( {\frac{1}{5}} \right) + \frac{4}{5}} \right] \cr & V = \pi \left[ { - 2 + 4\ln \left( 2 \right) + 2} \right] - \pi \left[ { - 5 + 4\ln \left( 5 \right) + \frac{4}{5}} \right] \cr & V = \pi \left[ {4\ln \left( 2 \right)} \right] - \pi \left[ {4\ln \left( 5 \right) - \frac{{21}}{5}} \right] \cr & V = \pi \left[ {4\ln \left( 2 \right) - 4\ln \left( 5 \right) + \frac{{21}}{5}} \right] \cr & V = \pi \left[ {4\ln \left( {\frac{2}{5}} \right) + \frac{{21}}{5}} \right] \cr & V = \left( {4\ln \left( {\frac{2}{5}} \right) + \frac{{21}}{5}} \right)\pi \cr} $$