 Hello and welcome to the session. Let us discuss the following question. It says find the area of the region bounded by the Curve x square is equal to 4y, y is equal to 2, y is equal to 4 and the yx is in the first quadrant Let us first understand how do we find the area of the region? bounded by the curve x is equal to f y, the yx is and the y coordinate y is equal to a and y equal to b is given by the integral a to b f y d y That is we integrate the function with respect to y over the limit a to b Which are the coordinates of y on the y-axis Which is equal to f of y Over a to b which is given by f b minus f a so this knowledge will Walk as key idea Let us now move on to the solution Here the curve given to us is x square is equal to 4y And it is a of the form x square is equal to 4 a y which is a parabola Y is equal to 2 and y is equal to 4 and we have to find the area of the region bounded by the curve Y is x square is equal to 4 y y equal to 2 y is equal to 4 So this is a parabola. This is the line y equal to 2 this is the line y is equal to 4 and we have to find the Area of the region in the first quadrant bounded by The curve and the lines that is this is the area which we have to find Now we are given that x square is equal to 4 y so x is equal to Plus minus root 2 y It's to root y and since we have to find area in just the first quadrant we ignore the negative sign and x is equal to To root y now we find the area denoted by a Y varies from 2 to 4 Fy is To root y d y Now we integrate this y to the power 1 by 2 d y Which is equal to 2 into y to the power 1 by 2 plus 1 upon 1 by 2 plus 1 and Y varies from 2 to 4 This is again equal to 2 into 2 by 3 Into y to the power 3 by 2 over 2 to 4 and this is equal to 4 by 3 into y to the power 3 by 2 but we substitute 4 becomes 4 to the power 3 by 2 minus 2 to the power 3 by 2 Now this can be again written as 4 into 4 to the power 3 by 2 can be written as 2 square to the power 3 by 2 Minus 4 into 2 to the power 3 by 2 can be written as 2 into 2 to the power 1 by 2 upon 3 This is again equal to 4 into 2 to the power 3 Minus 8 root 2 upon 3 Now 8 into 4 is 32 is 32 minus 8 root 2 upon 3 Hence the area of the region is 32 minus 8 root 2 upon 3 This completes the question and the session by for now take care have a good day