Question

Find the area of the region bounded by the parabola y = x 2 and

Answer 1

We have to find the area bounded by the parabola, x 2 =y and y=| x |. Draw a graph of these equations.

The figure above shows that the area bounded by the parabola and lines is symmetric about y-axis. So, the total area bounded by the curve and function is twice the area of OACO.

Solve the equation of the parabola, x 2 =y and y=x to find the point of intersection.

x 2 =x x 2 −x=0 x( x−1 )=0 x=0,1

The coordinates of point B are ( 1,0 ).

The area of the region OACO is,

Area of the region OACO=Area of the triangle OAB−Area of region OCABO

First, calculate the area of the triangle.

Area of triangle OAB= 1 2 ×1×1 = 1 2

To calculate the area of the region OCABO, we take a vertical strip in the region with infinitely small width as shown in the figure above.

To find the area of the region OCABO, integrate the area of the strip.

Area of region OCABO= ∫ 0 1 ydx (1)

The equation of the parabola is x 2 =y. From this equation find the value of y in terms of x and substitute in equation (1).

x 2 =y

Substitute x 2 for y in equation (1).

Area of the region OCABO= ∫ 0 1 x 2 dx = 1 3 [ x 3 ] 0 1 = 1 3

The area of the region OACO is,

Area of the region OACO= 1 2 − 1 3 = 3−2 6 = 1 6  sq units

Total area bounded by the parabola, x 2 =y and y=| x | is,

Total area=2×Area of the region OACO =2× 1 6 = 1 3  sq units

Thus, the area bounded by the parabola, x 2 =y and y=| x | is 1 3  sq units.