Question

Find the area of region bounded by the line $x=2$ and the parabola $y^2=8x$.

Answer 1

$\begin{array}{c}\frac{1}{1+x^2}-\log \left(1+x^2\right)\\ y^2=8x\\ y=\sqrt{8x}\\ y=2\sqrt{2x}\\ area=2\underset{0}{\overset{2}{\int }}ydx=2\underset{0}{\overset{2}{\int }}2\sqrt{2}\sqrt{x}dx\\ =4\sqrt{2}\underset{0}{\overset{2}{\int }}\sqrt{x}dx=4\sqrt{2}{\left[\frac{x^{3/2}}{3/2}\right]}_0^2\\ =\frac{8\sqrt{2}}{3}\left[2^{3/2}-0\right]=\frac{8\sqrt{2}}{3}2\sqrt{2}=\frac{32}{3}\end{array}$