Question

Find the area of the region bounded by the parabola $y^2=2x$ and the straight line $x-y=4$.

Answer 1

$y^2=2x$

$x=4+y$

$y^2=2(4+y)$

$y^2-2y-8=0$

$(y-4)(y+2)=0$

$y=4,-2$

So, $Area={\int }_{-2}^4(\frac{-y^2}{2}+4+y)dy$

$Area={\int }_{-2}^4(y-\frac{y^2}{2}+4)dy=(\frac{y^2}{2}-\frac{y^3}{6}+4y){\int }_{-2}^{4}Area=18squnits.$