 The session of numerical integration, this is Swati Nikam, Assistant Professor, Department of Humanities and Sciences, Valjan Institute of Technology, Soolapur. At the end of this session, student will be able to solve the numerical integration by using trapezoidal rule. Now for using this trapezoidal rule, friends we have to use first Newton-Court's formula. This is very basic and parent formula which is stated as integration x0 to xn ydx is equal to h times in bracket y0 into n plus n square by 2 into delta y0 plus half times bracket 2n cube minus 3n square upon 6 into delta square y0 plus 1 by 6 times n raised to 4 minus 4n cube plus 4n square whole divided by 4 into delta cube y0 plus and so on. Let us call this Newton-Court's formula as formula star. Now if we put n is equal to 1 in Newton-Court's formula, we get the required trapezoidal rule which we have to do for numerical integration. So the rule will be integration x0 to x0 plus nh ydx is equal to h by 2 into bracket y0 plus 2 times bracket y1 plus y2 plus and so on plus yn minus 1 plus yn. Now to summarize this formula, you can recall it as first term and last term are single. You can write them together and plus 2 times rest of the terms. Let us have a very important note with us. This is very important note for solving examples. Let us divide the interval x0 to xn in n equal parts with h is equal to b minus a whole divided by n. So division of this interval into n equal parts is done with the help of this width h which is given by this formula where b is our last ordinate xn, a is the first ordinate x0 and n is the total number of sub intervals that we have to do. Then the ordinate formed with this division will be x0, x1, x2, x3, x4 and so on up to xn. And they are given by x1 is equal to x0 plus h, x2 is equal to x0 plus 2h, then x3 is equal to x0 plus 3h and so on in this way if we continue xn will be x0 plus nh or in other words x1 is equal to x0 plus h, then x2 is equal to x1 plus h, then x3 is equal to x2 plus h and so on, xn is equal to xn minus 1 plus h. So see the next ordinate is obtained with the help of previous one. So x2 is obtained with the help of x1, x3 is obtained by x2 and so on. So that's why these ordinates are called as equi-spaced ordinates. Now friends please pause your video for a moment and divide the interval 0 to 10 into 10 equal parts. So if you do this type of division the interval 0 to 10 is divided into 10 equal parts as here the last ordinate b is 10, first ordinate a is 0 and we have to divide this into 10 equal parts so small n is equal to 10. So the formula I have already explained about h is the width h is b minus a upon n will become here 10 minus 0 divided by 10 which is equal to 1. Since our first ordinate x0 is 0 and x1 is obtained with this x0 as x1 is equal to x0 plus h which is equal to x0 value is 0, h value is 1 so 0 plus 1 is equal to 1. x2 can be obtained with the help of x1 as x2 is equal to x1 plus h that is x1 is 1, h is 1 so 1 plus 1 is equal to 2. Similarly x3 is x2 plus h that is 2 plus 1 is 3. We have to continue this process and at the last we will get the last ordinate x10 as x9 plus h is equal to 9 plus 1 is equal to 10. So in this way you have to write down the ordinate from the given interval. Now let us have an example so that we will see how can we use our numerical integration method that is trapezoidal rule particularly for this example. Now evaluate integration 0 to 2 e raise to x square dx taking 10 number of intervals. If friends if you see this integration carefully we cannot do this integration with the help of our analytic or calculus techniques directly. So for such type of integrations we have to take help of numerical methods especially numerical integration and here in this example we are going to use trapezoidal rule. So let me consider this integrand as y is equal to e raise to x square. Since we need to divide the interval into 10 number of intervals we have to consider small n is equal to 10. Therefore width h is equal to b minus a upon n our usual formula which I have already explained in earlier slide is equal to b is 2 here and a is 0. So 2 minus 0 upon 10 that is equal to 0.2. So h is 0.2 with this help of h we have to write down all the ordinate here. The values of x and y are tabulated as follows. So let's have all those values x0 to x10 in tabular format as so x0 is initial value 0, x1 as I have already explained x1 is x0 plus h that is 0 plus 0.2 is 0, 0.2. x2 is x1 plus h that is 0.2 plus 0.2 that is 0.4 it's very simple task in the same manner you have to get all the ordinate and finally x10 comes out to be 2. Now we need to go for all yi values with the help of this relation y is equal to e raise to x square. So for y0 it is e raise to x0 square that is e raise to 0 which is 1 therefore y0 value is 1. On the same line we will go for y1. So y1 is equal to e raise to x1 square that is e raise to 0.2 square which is 1.0408. On calculator you have to do this type of calculations and similarly we will go for y2, y3, y4, y5, y6, y7, y8, y9 and y10. Now we are going to use all these values in our formula. By using trapezoidal rule let me rewrite the trapezoidal rule for this example. So here we have terms from y0 to y10 therefore integration 0 to 2 e raise to x square dx is equal to h by 2 times in bracket first bracket as I have already told you first term plus last term y0 plus y10 plus 2 times all the rest terms. So here you have to write down from y2 to y9 addition of all the terms y2 plus y3 plus y4 plus y5 plus y6 plus y7 plus y8 plus y9. Now substitute the values from table over here we have all yi values here substituting all these values in the formula we will get it as h is 0.2 divided by 2 into bracket y0 y10 value plus 2 times rest of the values from our table. This calculation leads to the answer 17.0621 so this is the value of integration we have obtained with the help of numerical integration technique it is very simple rule to summarize and to calculate also. So for creation of this video I have referred a text book called as numerical method in engineering and science by Dr. B. S. Grival. Thank you and have a happy learning.