Find the area of the region bounded by the curve y= and the line y=x.

Given the boundaries of the area O befound are,


• Curve is y = √x


• Line y = x


Consider the curve


y2 = x


Substitute y = x


(x)2 = x


x2 –x = 0


x(x-1) =0


x = 1 (or) x = 0


substituting x in y = x


y = 1 (or) y = 0


So , the parabola meets the line y = √x at 2 points, A (1,1) and •(0,0)



As per the given boundaries,


• The parabola (y)2 = x, with vertex at O(0,0).


• Line y = x


The boundaries of the region to be found are,


Point A, where the curve (y)2 = x and the line y= x meet i.e. A (1,1)


Point O, the origin


Now, drop a perpendicular B on the x-axis from A, the point will be B(1,0)


Area of the required region = Area of OPAQO


Area of OPAQ•= Area under OPAB - Area under OQAB





[Using the formula ]



The Area of the required region


1