Chapter 13 - Multiple Integration - 13.2 Double Integrals over General Regions - 13.2 Exercises - Page 981: 18

$$8$$

$$\eqalign{ & \int_0^1 {\int_0^{2x} {15x{y^2}} dydx} \cr & {\text{Integrate with respect to }}y \cr & = \int_0^1 {\left[ {5x{y^3}} \right]_0^{2x}dx} \cr & = \int_0^1 {\left[ {5x{{\left( {2x} \right)}^3} - 5x{{\left( 0 \right)}^3}} \right]dx} \cr & = \int_0^1 {\left( {40{x^4} - 0} \right)dx} \cr & = 40\int_0^1 {{x^4}dx} \cr & {\text{Integrate with respect to }}x \cr & = \left[ {8{x^5}} \right]_0^1 \cr & {\text{Evaluating}} \cr & = 8{\left( 1 \right)^5} - 8{\left( 0 \right)^5} \cr & = 8 \cr} $$