Answer to Question #180267 in Calculus for Prathamesh

Question

Find area bounded by parabola y2 = 4ax and line y = mx.

Answer 1

The area enclosed between the parabola $y^2=4ax$ , and the line $y=mx$ , is represented by the shaded area$OABO$ as

The points of intersection of both the curves are$(0,0)$ and$(\dfrac{4a}{m^2},\dfrac{4a}{m} )$ .

We draw AC perpendicular to $x-axis$ .

So, $Area OABO= Area OCABO - Area (OCA)$

$= \int_0^{\dfrac{4a}{m^2}}2\sqrt{ax}-$ $\int_0^{\dfrac{4a}{m^2}}mxdx$

=$2\sqrt{a}[\dfrac{x^{1.5}}{1.5}]_0^\dfrac{4a}{m^2}$ $-$ $m[\dfrac{x^2}{2}]^{\dfrac{4a}{m^2}}_0$

$=\dfrac{8a^2}{3m^3} units$