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Question

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

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Solution

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( x1 )=0 x=0,1

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

The area of the region OACO is,

AreaoftheregionOACO=AreaofthetriangleOABAreaofregionOCABO

First, calculate the area of the triangle.

AreaoftriangleOAB= 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.

AreaofregionOCABO= 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).

AreaoftheregionOCABO= 0 1 x 2 dx = 1 3 [ x 3 ] 0 1 = 1 3

The area of the region OACO is,

AreaoftheregionOACO= 1 2 1 3 = 32 6 = 1 6 squnits

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

Totalarea=2×AreaoftheregionOACO =2× 1 6 = 1 3 squnits

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


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