3.6.1

[STARTS AT 13:16]

Instructor: So let's take the same procedure-- the same steps we have done in the last few. Let's first find delta x. Delta x would be a and b or 0 and 1 respectively. So that is 1 minus 0 or 1 divided by four subintervals. N equals 4. 

All right, so a is going to be 0. I'm going to label it x0 as well. x1 would be 1/4. x2 would be 1/2. And hopefully, this looks somewhat familiar. x4 is 1, which is also our b term. Now, we're going to find f of x0, f of x1, f of x2, f of x3, f of x4, and we'll find that is 0. We're squaring these. 1/16, 1/4, 9/16, and 1. 

Now, how this differs from the midpoint rule is in the midpoint rule, we average the x values and then evaluate them. With the trapezoidal rule, we evaluate them and then we average them. We're going to average each of these values. So that would give us a 1/32, a 5/32, 13/32, and I'm not this fast with arithmetic. I have worked this out previously. 

So I suggest that you do check these even though I have checked them repeatedly. So each of these are the f of-- that's f of x0 plus f of x1 divided by 2. That's what that is. This is f of x1 plus f of x2 divided by 2. f of x2 and f of x3 divided by 2. And this is f of x3 plus f of x4 divided by 2. And as such, our formula says we should add these, or at least my alternative formula. 

So if I add all of those up, I would get the sum of i equals 0 to n minus 1 of f of xi plus f of xi plus 1 over 2. That itself would be 44/32. 

Now, I need to multiply by 1/4. Yes, that was our delta x. Say this x, which is 1/4 so that T4 is going to be 11/32, which is approximately 0.34375.

[ENDS AT 16:47]