# Upper and lower sums

1. Nov 21, 2005

### sukisyo

In an equation, the upper sum is Mi = 0+i(2/n)
and the lower sum is mi = 0+(i-1)(2/n)
So the question is why is it (i-1) for the lower sum and only i for the upper sum?

Any help is highly appreciated! ^_^

2. Nov 21, 2005

### BerkMath

State the entire problem.

3. Nov 21, 2005

### sukisyo

it's not a problem, but a question.

Set in the interval [0,2], it asks me to explain why I need to have i minus 1 in finding the lower sum (the left endpoints) where as in finding the upper sum, it is just i.

4. Nov 21, 2005

### BerkMath

Lower/Upper sum of what?

5. Nov 21, 2005

### sukisyo

ok. here is everything. i don't know if it helps.

Find the upper and lower sum for the region bounded by the graphy of f(x) = x^2 and x-axis between x=0 and x=2. To begin, partition the interval [0,2] into n sublevels, each of length (triangle X) = (b-a)/n = (2-0)/n = 2/n

Left endpoints:
https://www.physicsforums.com/latex_images/39/394830-1.png [Broken]
Right endpoints:
https://www.physicsforums.com/latex_images/39/394830-3.png [Broken]

QUESTION: why does the equation need (i-1) for finding the left endpoints when it only needs i in finding the right endpoints?

Last edited by a moderator: May 2, 2017
6. Nov 21, 2005

### mathphys

Hi

First, those are the positions of the left/right endpoints, not the upper/lower sums.

What happens when i=1 and i=n? (remeber that i=1,2,...n)

Last edited: Nov 21, 2005
7. Nov 21, 2005

### sukisyo

I tried to understand what you are asking me but i'm getting confused... again.
So i asked a friend and she said:

"Mi and mi are the right and left endpoints so to find its exact value, we take the i (which names the specific interval) and multiply it by the value of the subintervals (delta x)and since the area of the subinterval is height times width, then height is found by f(mi) or f(Mi) and width is delta x. mi is (i-1) because lets say Mi is at i
then the left endpoint (mi) is the right endpoint of the previous interval so that is why it is i-1."

She isn't really sure about her answer but i can't think of anything else so i'm just going to accept it. Also if this is the answer, it's like common sense, so i'm going to be feeling really stupid.

Thanks for all your help though. I really appreciate your time and hard effort in attemping to free me from my dilemma. ^_^

8. Nov 21, 2005

### HallsofIvy

Staff Emeritus
You finally told us that the function in question was x2!

That is an increasing function. If you draw a horizontal line at xi-1 and xi which is lower? Which gives a rectangle that is smaller?