Find area of region

Given,


• R = {(x,y): x2 ≤ y ≤ x}


From the set we have the curve, y = x2 ------ (1)


Also the line equation y = x ------ (2)



As per the given boundaries,


• The curve y =x2, has only the positive numbers as x has even power, so it is about the y-axis equally distributed on both sides.


• y = x is a line passing through the origin.


The boundaries of the region to be found are,


Point A, where the curve y = x2 and y=x meet, i.e. A (1,1)


Point O, which is the origin


Drop a perpendicular D on the x-axis from A, where D = (1,0)


Now,


Area of the required region = Area of OPAQO.


Area of OPAQ•= Area of OPAD•– Area of OQADO





[Using the formula ]



The Area of the required region


1