 Hi friends, I am Purva and today I will help you with the following question. Choose the correct answer in the following and the question is area lying between the curves y square is equal to 4x and y is equal to 2x is a 2 upon 3 b 1 upon 3 c 1 upon 4 d 3 upon 4 Let us now begin with the solution. Now we have to find the area enclosed between the curve y square is equal to 4x and y is equal to 2x so we have equation of curves y square is equal to 4x and let us mark this as equation 1 and we have y is equal to 2x and we mark this as equation 2. Now to find the points of intersection of these two equations what we do is we put the value of y from equation 2 in equation 1. So from 1 and 2 we have we put the value of y that is 2x in equation 1. So we get 2x whole square is equal to 4x and this implies now 2x whole square gives 4x square minus 4x is equal to 0 and this further implies taking the 4x common we get 4x into x minus 1 is equal to 0 and this implies either x is equal to 0 or x is equal to 1. Now if x is equal to 0 then putting the value of x in equation 2 we get y is equal to 0 and if x is equal to 1 then again putting the value of x in equation 2 we get y is equal to 2. Therefore the two points of intersection are 0, 0 and 1, 2. Now y square is equal to 4x is the equation of parabola whose vertex is 0, 0 and it is symmetric about x axis. So this is the parabola y square is equal to 4x and y is equal to 2x is a line passing through 0, 0 and 1, 2. So this is the line y is equal to 2x which passes through 0, 0 and 1, 2. And this shaded region is the region whose area is to be found out. Therefore we have area of shaded region is equal to area of region OPAP minus area of triangle OAP. Therefore we have area of shaded region is equal to area of region OPAP minus area of triangle OAP. Now area of this whole region OPAP is given by here limit is from 0 to 1 and equation of parabola is y square is equal to 4x so we get y is equal to 2 root x. So we get area of region OPAP is equal to integral limit is from 0 to 1 2 root x dx minus now area of triangle OAP is given by now again this region lies between 0 and 1 and equation of line is y is equal to 2x so we get area of triangle OAP is equal to integral limit is from 0 to 1 2x dx. And this is equal to 2 into now integration of root x is equal to x raised to the power 3 by 2 upon 3 by 2 and here limit is from 0 to 1 minus 2 into now integration of x is equal to x square upon 2 and here again limit is from 0 to 1 and this is equal to 2 into 2 upon 3 into now putting the limits we get putting the upper limit 1 in place of x we get 1 to the power 3 by 2 which is equal to 1 minus putting lower limit 0 in place of x we get 0 minus 2 into 1 upon 2 into again putting the limits we get putting upper limit 1 in place of x we get 1 minus putting lower limit 0 in place of x we get 0 and this is equal to now 2 into 2 upon 3 is equal to 4 upon 3 into 1 minus 0 is equal to 1 minus now cancelling of 2 here we get 1 and 1 minus 0 is again equal to 1 so we get 1 into 1 and this is equal to now 4 upon 3 into 1 is equal to 4 upon 3 minus 1 into 1 gives 1 and this is equal to now 4 upon 3 minus 1 gives 1 upon 3 and this is same as option B so we get the correct answer is B thus we write our answer as B hope you have understood the solution bye and take care