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Question

Area of the region bounded by the curve y 2 = 4 x , y -axis and the line y = 3 is A. 2 B. C. D.

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Solution

We have to find the area bounded by the parabola y 2 =4x and line y=3. Draw the graph of these equations.



Figure (1)

Solve the equations to find the point of intersection.

y 2 =4x ( 3 ) 2 =4x x= 9 4

To find the area of OABO, we take a horizontal strip in the region with infinitely small width dy and integrate for the area of the strip.

AreaoftheregionOABO= 0 3 xdy (1)

From the equation of parabola, find the value of x in terms of y and substitute in equation (1).

y 2 =4x x= y 2 4

Substitute y 2 4 for x in equation (1) and integrate.

AreaoftheregionOABO= 0 3 y 2 4 dy = 1 4 0 3 y 2 dy = 1 4 [ y 3 3 ] 0 3 = 1 4 [ 3 3 3 0 3 3 ]

Simplify further,

AreaoftheregionOABO= 1 4 [ 27 3 ] = 1 4 [ 9 ] = 9 4 squnits

The area calculated for the region OABO is 9 4 squnits.

Thus, out of all the four options, option (B) is correct.


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