Upper & Lower Sums: Why (i-1) vs i?

  • Thread starter sukisyo
  • Start date
  • Tags
    Sums
In summary, the question is asking why for the lower sum (left endpoints), the equation uses (i-1) and for the upper sum (right endpoints), it only uses i. The answer is that for the lower sum, the left endpoint is the right endpoint of the previous interval, so it is i-1. This is because the function x^2 is increasing, meaning the rectangle formed by the horizontal line at xi-1 and xi is smaller than the one formed by the horizontal line at xi and xi+1.
  • #1
sukisyo
4
0
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! ^_^
 
Physics news on Phys.org
  • #2
State the entire problem.
 
  • #3
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
Lower/Upper sum of what?
 
  • #5
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:
  • #6
Hi

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

Now, think about this, is 0 a left or a right endpoint? and 2?
What happens when i=1 and i=n? (remeber that i=1,2,...n)
 
Last edited:
  • #7
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 let's 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
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?
 

1. What is the difference between upper and lower sums?

Upper and lower sums are mathematical concepts used in calculus to approximate the area under a curve. The main difference between them is that upper sums use the maximum value of the function within each subinterval to calculate the area, while lower sums use the minimum value.

2. Why is (i-1) used instead of i in upper and lower sums?

The (i-1) and i in upper and lower sums refer to the subintervals used to approximate the area under the curve. (i-1) is used because it represents the left endpoint of each subinterval, while i represents the right endpoint. This allows for a more accurate approximation of the area under the curve.

3. How are upper and lower sums calculated?

To calculate upper and lower sums, you first divide the interval of the function into smaller subintervals. Then, you find the maximum and minimum values of the function within each subinterval and multiply it by the width of the subinterval. Finally, you add up all the resulting products to get the approximate area under the curve.

4. When should I use upper or lower sums?

Upper and lower sums are used in calculus to approximate the area under a curve when an exact solution is not possible. The choice between using upper or lower sums depends on the function being evaluated and the level of precision required. In general, upper sums tend to overestimate the area while lower sums tend to underestimate it, so using both can provide a better approximation.

5. What is the significance of (i-1) vs i in upper and lower sums?

The use of (i-1) and i in upper and lower sums is important because it allows for a more accurate approximation of the area under the curve. By using the left and right endpoints of each subinterval, we can capture more of the variation in the function and reduce the error in the approximation.

Similar threads

  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
419
  • Calculus and Beyond Homework Help
Replies
1
Views
96
  • Calculus and Beyond Homework Help
Replies
16
Views
475
  • Calculus and Beyond Homework Help
Replies
5
Views
490
  • Calculus and Beyond Homework Help
Replies
1
Views
767
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
653
  • Calculus and Beyond Homework Help
Replies
3
Views
355
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
Back
Top