Q. 15.0( 4 Votes )

Find the area of the region bounded by the curves y2 = 9x, y = 3x.

Answer :

y2 = 9x is the equation of a parabola and y = 3x is a straight line passing through origin


let us first roughly draw those equations


In y2 = 9x negative values of x are not allowed hence the graph is on right of X-axis that is the parabola opening to the right



To find point of intersection solve both the equations simultaneously


Put y = 3x in y2 = 9x


(3x)2 = 9x


9x2 = 9x


x2 – x = 0


x(x – 1) = 0


x = 0 and x = 1


Put x = 1 and x = 0 in y = 3x we will get y = 3 and y = 0 respectively hence the parabola and straight line intersect at (1, 3) and (0, 0)


We have to find the area between parabola and straight line


That area will be the area under parabola minus the area under the straight line from x = 0 to x = 1 as shown in figure below


area between parabola and straight line = area under parabola – area under straight line …(i)



Let us calculate the area under parabola


y2 = 9x


y = 3√x


Integrate from 0 to 1









Now let us calculate area under the line y = 3x that is area of triangle OAB


y = 3x


Integrate from 0 to 1






Using (i)


area between parabola and straight line = 2 – = 1/2 unit2


Hence area of the region bounded by the curves y2 = 9x and y = 3x is 1/2 unit2


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