What is the area of the region enclosed between the curve y^2=2 x and the straight line y=x ?

Step-by-step solution

Two given graphs are y^2=2 x and y=x . Point of intersections are : aligned & x^2=2 x x^2-2 x=0 & x(x-2)=0 & x=0 and 2 & y=0 and 2 aligned So, points of intersections are (0,0) and (2,2) . The graph are The required area is the shaded portion A= | ₀^2 (y₁-y₂ ) d x | aligned & = | ₀^2( 2 x-x) d x |= [ 2 2 (x)^ 3 / 2 3 - x^2 2 ]₀^2 & = 2 2 3 (2)^ 3 2 -2= 8 3 -2= 2 3 aligned