Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis in the first quadrant.

The equation of curve is y² = x
Required area = 
                        

 

The equation of parabola is y² = x
From the given condition, area OAP under y² = x between x = 0 and x = a = area ABQP under y² = x between x = a and x = 4.

The equation of parabola is
y² = 4 x
The equation of line is
x = 3
Also, we know that parabola is symmetric about x-axis
∴ required area = 2 (area ORPO)

The equation of curve is x² = 4y, which is an upward parabola.
Lines are y = 2 and y  = 4
Required area  = Area ABCD