Mathematics - Area Under Curves Question with Solution | TestHub

Refer figure Required area is shaded. Point of intersection of given curves are ${\mathrm{y}}^2=8\mathrm{x}$ and ${\mathrm{x}}^2=8\mathrm{y}$ i.e. $$\begin{array}{rl}& {\left(\frac{x^2}{8}\right)}^2=8x\Rightarrow x\left(x^3-512\right)=0\\ & \therefore O\equiv (0,0)\text{ and }P\equiv (8,8)\\ & A={\int }_0^8(2\sqrt{2})(\sqrt{x})dx-{\int }_0^8\frac{x^2}{8}dx\\ & {=\frac{2\sqrt{2}}{\left(\frac{3}{2}\right)}{x}^{\frac{3}{2}}]}_0^8-\frac{1}{24}{\left[x^3\right]}_0^8=\left(\frac{4\sqrt{2}}{3}\right)(8\sqrt{8})-\frac{1}{24}(512)=\frac{64}{3}\text{ sq. units }\end{array}$$