00:01
Hello, let's look at the question we have been given to use the trapezoidal rule, midpoint rule and simpson's rule to approximate the integral with the specified value of n from 0 to 3, 1 by 4 plus y to the power 5 for dy when our n is equal to 6.
00:18
You see here we write this fy is equal to 1 by 4 plus y to the power 5.
00:24
Now we take delta y is equal to b minus a by n is equal to 3 minus 0 by 6 which is 1 by 2.
00:35
So our y0 is equal to 0, y1 is equal to 1 by 2, y2 is equal to 1, y3 is equal to 3 by 2, y4 is equal to 2, y5 is equal to 5 by 2 and our y6 is equal to 3.
00:56
For first factor use the trapezoidal rule.
01:04
For a and b, our fy, dy is equal to delta y by 2 fy0 plus 2 fy1 plus 2 fy2 plus up to which is equal to 0 .349477.
02:00
Now we use the midpoint rule.
02:10
We take from a to b, fy, dy is equal to delta y f m1 plus f plus up to f mn.
02:30
Now we will find m1, m2, m3 from the values that we got here.
02:35
From here we can see that our m1 will be 0 plus 1 by 2 by 2 that is 1 by 4.
02:45
We can write down here our m1 will be 1 by 4.
02:49
Similarly we can find our m2 that will be 3 by 4, m3 that will be 5 by 4, m4 that will be 7 by 4, m5 that will be 9 by 4 and m6 that will be 11 by 4.
03:06
For example we have got these values by adding these values as we added y1 and y2 here and got m1...