Question

I have the answer to number 3, but need help on 4 and 5. I will up-vote.

The way we approximate things, we make sure things are evenly spaced. Shifting things left and right doesn't affect an integral, since it's an area We will worry about horizontal scaling at the end. So let's assume (without loss of generality) that the r values of our coordinates are 0,1,2, and 3. We will write our points as (0, fo), (1, fi), (2, f2), and (3,fs). 3. The general form of a cubic is Ar3 +B2 CD. Set up a system of equations and solve for A, B, C, and D is terms of fo, fı, f2, and fs. THIS IS ALGEBRAICALLY INTENSIVE! 4. What is the integral of the cubic between 0 and 3, in terms of fo, f f2, and f3 5. Now we need to worry about our horizontal scaling. If we scale hori- zontally by a factor of Ar, what do we do to the integral above? Noo, obkein value

Answer 1

f(x) = Ar' + Br' + Cr + D

0 0 9 81

Now substitute the values of A,B,C,D from your calculation and get the result in terms of .

fo, fi, f2andfa

If we have a horizontal scaling by $\Delta x$ the integral is taken from 0 to $3\Delta x$ instead of 0 to 3.

So it becomes

81