 Hello and welcome to the session. In this session we will discuss about the area between two curves. Here we will find out the area of the region enclosed between two curves y equal to fx and y equal to gx and the lines x equal to a, x equal to b. This is the curve y equal to fx and this curve is y equal to gx. This is the line x equal to a and this is the line x equal to b. Now the area enclosed between these two curves and the lines x equal to a and x equal to b is this shaded portion. So we are supposed to find the area of this shaded portion and in this case we have fx is greater than equal to gx in the closed interval a, b. Consider this vertical strip of height fx minus gx and the width of this strip is dx. So the elementary area dA of this strip is given by fx minus gx dx and the total area is the result of adding up the elementary areas of thin strips across this area which would be given by a equal to integral a to b dA. This is equal to integral a to b fx minus gx dx. So this is the area a of the region enclosed between two curves y equal to fx, y equal to gx and the lines x equal to a, x equal to b. And in this case we have fx is greater than equal to gx in the interval a, b. If suppose we have the curve fx is greater than equal to curve gx in the closed interval a, c and the curve fx is less than equal to the curve gx in the closed interval c, b where the c is a point greater than a and less than b that is it lies between a and b. The c is a point between a and b. This curve fx is greater than equal to gx in the interval a, c and gx is greater than equal to fx in the interval c, b. Now the area of the regions bounded by the curves is the shaded portion so this total area is equal to area of the region a, c, b, d, a plus area of the region b, p, r, q, b. Now area of this region a, c, b, d, a would be equal to integral a to c fx minus gx dx plus now the area of the region b, p, r, q, b is equal to integral c to b dx minus fx dx. Let's try and find out the area of the region bounded by the two parabolas y equal to x square and y square equal to x. This is the parabola y equal to x square and this is the parabola x equal to y square. Now the point of intersection of these two parabolas is this point o which coordinates 0, 0 and this point b, a which coordinates 1, 1. We need to find the area bounded by these two parabolas that is this shaded portion would be the area bounded by these two parabolas. Our one curve is y square equal to x this gives us y equal to root x let this be equal to fx so this is our one curve then the other curve is y equal to x square let this be equal to gx this is our second curve. Now here we have fx is greater than equal to gx in the interval 0, 1. Now the area of the shaded portion that is the required area would be equal to integral 0 to 1 fx minus gx dx that is equal to integral 0 to 1. Now our fx is square root x minus gx is x square dx. So this is equal to 2 upon 3 x to the power 3 upon 2 minus x cube upon 3 which goes from the limits 0 to 1 this is further equal to 2 upon 3 minus 1 upon 3 which is equal to 1 upon 3. So we have 1 upon 3 square units is the area of this shaded portion that is the area bounded by the two parabolas x equal to y square and y equal to x square. This completes the session hope you have understood how do we find the area bounded by the two curves.