Chapter 13 - Multiple Integration - 13.2 Double Integrals over General Regions - 13.2 Exercises - Page 982: 69

\[V=\frac{2}{3}\]

\[\begin{align} & \text{Let the region }R=\left\{ \left( x,y \right):0\le y\le 1-x,\text{ 0}\le x\le 1\text{ } \right\} \\ & \text{The volume is given by} \\ & V=\int_{0}^{1}{\int_{0}^{1-x}{\left[ \left( 2-{{x}^{2}}-{{y}^{2}} \right)-\left( {{x}^{2}}+{{y}^{2}} \right) \right]}}dydx \\ & V=\int_{0}^{1}{\int_{0}^{1-x}{\left( 2-{{x}^{2}}-{{y}^{2}}-{{x}^{2}}-{{y}^{2}} \right)}}dydx \\ & V=\int_{0}^{1}{\int_{0}^{1-x}{\left( 2-2{{x}^{2}}-2{{y}^{2}} \right)}}dydx \\ & \text{Integrating} \\ & V=\int_{0}^{1}{\left[ 2y-2{{x}^{2}}y-\frac{2{{y}^{3}}}{3} \right]_{0}^{1-x}}dx \\ & V=\int_{0}^{1}{\left[ 2\left( 1-x \right)-2{{x}^{2}}\left( 1-x \right)-\frac{2{{\left( 1-x \right)}^{3}}}{3} \right]}dx \\ & V=\int_{0}^{1}{\left[ 2-2x-2{{x}^{2}}+2{{x}^{3}}-\frac{2{{\left( 1-x \right)}^{3}}}{3} \right]}dx \\ & \text{Integrating} \\ & V=\left[ 2x-{{x}^{2}}-\frac{2}{3}{{x}^{3}}+\frac{2{{x}^{4}}}{4}+\frac{2{{\left( 1-x \right)}^{4}}}{12} \right]_{0}^{1} \\ & V=2\left( 1 \right)-{{\left( 1 \right)}^{2}}-\frac{2}{3}{{\left( 1 \right)}^{3}}+\frac{2{{\left( 1 \right)}^{4}}}{4}+\frac{2{{\left( 1-1 \right)}^{4}}}{12}-\frac{2{{\left( 1-0 \right)}^{4}}}{12} \\ & V=2\left( 1 \right)-{{\left( 1 \right)}^{2}}-\frac{2}{3}{{\left( 1 \right)}^{3}}+\frac{2{{\left( 1 \right)}^{4}}}{4}+\frac{2{{\left( 1-1 \right)}^{4}}}{12}-\frac{2{{\left( 1-0 \right)}^{4}}}{12} \\ & V=\frac{2}{3} \\ \end{align}\]