Question

Evaluate the integral, I  (4x  2y) dxdy 3 in the region R bounded by x  0, x  2, y  x and y  x  2 .

Answer 1

$$\displaystyle\int_{0}^{2}\displaystyle\int_{x}^{x^2}4x^2ydydx=\displaystyle\int_{0}^{2}[2x^2y^2]\begin{matrix} x^2 \\ x \end{matrix}dx$$

$$=2\displaystyle\int_{0}^{2}(x^6-x^4)dx=2\big[\dfrac{x^7}{7}-\dfrac{x^5}{5}\big]\begin{matrix} 2 \\ 0 \end{matrix}$$

$$=2(\dfrac{2^7}{7}-\dfrac{2^5}{5})=\dfrac{64}{35}(20-7)=\dfrac{832}{35}$$