Riemann Sums

[maccaman]

I have just been doing some Riemann Sums, but i must find the midpoint. Now i am doing some research on how to do it but i dont get the formula.

Function i am using is y=x^2, over x=2 and x=12, with n=20 subintervals.

For the left sum i used something like this
1/2 * (2^2 + 2.5^2 + 3^2 + 3.5^2 + ........ + 11.5^2)

now for the right i used somethign like this
1/2 * (2.5^2 + 3^2 + 3.5^2 + ......... + 12^2)

From my basic understanding, i am a highschool student, all i do is this
LEFT (a-b / n) multiplied by (a^2 + ..... (every 0.5^2 interval until
11.5^2) (miss out 12 ^2) ).

RIGHT i did the same for right, with the exception of leaving out a^2, and put in b^2 instead.

if someone could help me a bit on the MIDPOINT one would be great, i know the sum, just want to know how to do it.

P.S. Ive seen upper and lower, is this the same as right and left. Thanks for any help

[matt grime]

I'm slightly confused as to what level you're at here.

It looks like you're doing a numerical integral using rectangles whose heights are given by the function values at the left end of each interval, the right end of each interval, and (for the midpoint) the value at the midpoint of each interval.

so here you'd need 2.25^2, 2.75^2, 3.25^2 etc as the values.


The Upper and Lower Riemann Sums of a partition are not (necessarily) these. The lower sum is where you use the minimal value of the function in each interval to define a rectangle, and the upper is where you define the rectangle with the largest value in the interval. It so happens in the case of x^2 that the minimal value and maximal value occur at the end points of each interval.

Looking ahead to college - a function is integrable when these Upper and lower sums are 'well behaved': what is the difference between your left and right end point sums? suppose you pick a finer partition, ie with more subintervals, will the difference between left and right sums get smaller? If i give you some error term e (a small positive number) can you pick small enough intervals so that the upper and lower sums are less than e apart? (The answer is yes, and that's what it means for the function to be Riemann integrable).

[maccaman]

How do i find the midpoint from this formula, given from the textbook with that above function:

n
(sigma) f(mi) (delta x)
I=1

where mi = X i-1 + X i
2

How the hell do i do this. Please help

[matt grime]

I already said what the midpoints were.

Let's derive them again:


What is the mid point of the first subinterval?

The subinterval is from 2 to 2.5

so (2+2.5)/2 = 2.25

the second subinterval si 2.5 to 3, so the midpoint is

(2.5+3)/2 = 2.75

the midpoint of each subinterval is the point half way along the interval.

so you need to add up 2.25^2, 2.75^2,...