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