 Hello and welcome to the session I am Deepika here. Let's discuss the question which says find the area of the circle 4x square plus 4y square is equal to 9 which is interior to the parabola x square is equal to 4y. Let us first understand how to find the area between two curves. Suppose we are given two curves that is if y is equal to fx and y is equal to gx are two curves then the area between them is equal to a and x is equal to b is given by integral a to b f of x minus g of x dx where fx is greater than equal to gx in the closed interval a b. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. First of all we will identify the region whose area we have to find that is we have to find the area of the circle 4x square plus 4y square is equal to 9 which is interior to the parabola x square is equal to 4y. Now we have plus 4y square is equal to 9 this implies x square plus y square is equal to 9 over 4 and this implies x square plus y square is equal to 3 over 2 square. Thus this is a circle with center origin that is with center 0 0 and radius equal to 3 over 2 is equal to 4y is a parabola with vertex origin that is the coordinates of the vertex are 0 0 and symmetrical about y axis. Now we will find the points of intersection of the given two curves. Now 4y implies y is equal to x square upon 4 restituting the value of y in the equation x square plus y square is equal to 9 over 4 we get plus x is to power 4 over 16 is equal to 9 over 4 this implies 16 x square plus x is to power 4 is equal to 9 over 4 into 16. This implies x is to power 4 plus 16 x square minus 36 is equal to 0 now we will factorize this equation this implies x is to power 4 plus 18 x square minus 2 x square minus 36 is equal to 0 and this implies x square into x square plus 18 minus 2 into x square plus 18 is equal to 0 this implies x square minus 2 into x square plus 18 is equal to 0 and this implies either x square minus 2 is equal to 0 or x square plus 18 is equal to 0 this implies either x is equal to plus minus under root of 2 or is equal to minus 18 which is not possible. x is equal to plus under root 2 then y is equal to root 2 square upon 4 which is equal to 1 over 2 again when x is equal to minus under root of 2 then y is equal to minus under root of 2 square over 4. which is again equal to 1 over 2 hence the points of intersection are root 2 1 by 2 minus root 2 1 by 2 so this is root 2 1 by 2 minus root 2 1 by 2. so we have to find the area of this shaded region so the required area is equal to 2 into area of the shaded region in the first quadrant and this is equal to 2 into integral 0 to root 2. y dx of circle that is minus y dx of parabola and this is equal to 2 into integral from 0 to root 2 now from the equation of circle we have y is equal to under root of 9 by 4 minus x square dx minus 0 to root 2. now from the equation of parabola we have y is equal to x square upon 4 dx in this part we will use the formula integral a square minus x square dx is equal to x over 2 under root of a square minus x square plus a square upon 2 sin inverse x upon a plus c sin inverse x over 2. so this is equal to 2 into x by 2 into under root of 9 by 4 minus x square plus a square upon 2 which is 9 by 8 sin inverse x upon a that is 2 x upon 3 and the limits are from 0 to root 2 minus 1 by 4. and this is equal to 2 into root 2 over 2 into under root of 9 by 4 minus root 2 square is 2 plus 9 by 8 sin inverse 2 root 2 over 3 upon 2 into under root 2. this is equal to root of 9 by 4 minus 0 which is 0 plus 9 by 8 plus 1 by 4 into root 2 cube this is equal to 2 root 2 over 2 into 1 by 2 plus 2 into 9 over 8 sin inverse 2 root 2 over 3 minus 2 into 2 root 2 over 4 into 3. and this is equal to root 2 over 2 minus plus 9 over 8 which is 9 by 4 sin inverse 2 root 2 over and this is equal to 3 root 2 minus 1 by 6 plus 9 over 4 sin inverse 2 root and this is equal to root 2 over 6 plus 9 by 4 sin inverse 2 root 2 over 3 hence the required area is equal to root 2 over 6 plus 9 by 4 sin inverse 2 root 2 over 3. so the answer for the above question is root 2 over 6 plus 9 by 4 sin inverse 2 root 2 over 3 I hope the solution is clear to you bye and take care.