Chapter 13 - Multiple Integration - 13.2 Double Integrals over General Regions - 13.2 Exercises - Page 982: 67

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

\[\begin{align} & \int_{0}^{\sqrt[3]{\pi }}{\int_{y}^{\sqrt[3]{\pi }}{{{x}^{4}}\cos \left( {{x}^{2}}y \right)}dxdy} \\ & \text{Using the graph to switch the order of integration} \\ & R=\left\{ \left( x,y \right):0\le y\le x,\text{ 0}\le x\le \sqrt[3]{\pi }\text{ } \right\} \\ & \text{Then}, \\ & \int_{0}^{\sqrt[3]{\pi }}{\int_{y}^{\sqrt[3]{\pi }}{{{x}^{4}}\cos \left( {{x}^{2}}y \right)}dxdy}=\int_{0}^{\sqrt[3]{\pi }}{\int_{0}^{x}{{{x}^{4}}\cos \left( {{x}^{2}}y \right)}dydx} \\ & =\int_{0}^{\sqrt[3]{\pi }}{{{x}^{4}}\int_{0}^{x}{\cos \left( {{x}^{2}}y \right)}dydx} \\ & \text{Integrating} \\ & =\int_{0}^{\sqrt[3]{\pi }}{{{x}^{4}}\left[ \frac{1}{{{x}^{2}}}\sin \left( {{x}^{2}}y \right) \right]_{0}^{x}dx} \\ & =\int_{0}^{\sqrt[3]{\pi }}{\left[ {{x}^{2}}\sin \left( {{x}^{2}}y \right) \right]_{0}^{x}dx} \\ & =\int_{0}^{\sqrt[3]{\pi }}{\left[ {{x}^{2}}\sin \left( {{x}^{2}}x \right)-{{x}^{2}}\sin \left( 0 \right) \right]dx} \\ & =\int_{0}^{\sqrt[3]{\pi }}{{{x}^{2}}\sin \left( {{x}^{3}} \right)dx} \\ & =-\frac{1}{3}\left[ \cos \left( {{x}^{3}} \right) \right]_{0}^{\sqrt[3]{\pi }} \\ & =-\frac{1}{3}\left[ \cos {{\left( \sqrt[3]{\pi } \right)}^{3}}-\cos {{\left( 0 \right)}^{3}} \right] \\ & =-\frac{1}{3}\left[ \cos \left( \pi \right)-\cos \left( 0 \right) \right] \\ & =-\frac{1}{3}\left[ -1-1 \right] \\ & =\frac{2}{3} \\ \end{align}\]