 Hello friends, let's discuss the following question. It says find the area of the region bounded by Y square is equal to 9x, x is equal to 2, x is equal to 4 and the xx is in the first quadrant. Let us first understand how do we find area of the region bounded by the curve? Area of the region bounded by the curve Y equal to fx and the xx is and the coordinates x is equal to a, x is equal to b is given by integral a to b fx dx. So this will be the key idea. Let us now move on to the solution. The curve given to us is y square is equal to 9x and it is of the form y square is equal to 4ax, which is actually the parabola. We have to find the area bounded by the curve y square is equal to 9x, x is equal to 2 and x is equal to 4. y square is equal to 9x is a parabola and this is the line x is equal to 2. So we have to find area under the curve y square is equal to 9x. The line x is equal to 2 and x is equal to 4. So this is the region for which we have to find the area and we have to find the area in the first quadrant. Now y square is equal to 9x. So y is equal to root 3x, it is 3 root x, square root of 9 is 3, it is root x, it is plus minus. But since we have to find the area in the first quadrant, we ignore the negative sign and we will find the area of the region under the curve 3 root x, x is equal to 2 and x is equal to 4. So the area denoted by a is given by the integral 2 to 4 3 root x dx which is equal to integral 2 to 4 root x dx which is equal to 3 into integral of x to the power 1 by 2 is 1 by 2 plus 1 upon 1 by 2 plus 1. And this is why the formula for the integral of x to the power n dx it is given by x to the power n plus 1. Upon n plus 1 this becomes 3 into x to the power 3 by 2 upon 3 by 2 which is equal to 2 into x to the power 3 by 2 and this is under the region x is equal to 2 to x is equal to 4 and this is equal to 2 into 4 to the power 3 by 2 upper limit minus the lower limit minus 2 to the power 3 by 2. This is again equal to 2 into 4 can be written as 2 square to the power 3 by 2 minus 2 into 2 to the power 3 by 2. This is again equal to 2 into 2 to the power 3 minus 2 into 2 to the power 3 by 2 can be written as 2 into 2 to the power 1 by 2. This is equal to 16 minus 4 root 2. And this is why the second fundamental theorem which says that integral a to b fx dx which is equal to sum fx is which is given by fb minus fa. Area of the region bounded by the curve y square is equal to 9x the line x is equal to 2 and x is equal to 4 is 16 minus 4 root 2. This completes the question and the session. I will now take care and have a good day.