| Thread Closed |
finding upper and lower sums of a region |
Share Thread | Thread Tools |
| Dec6-04, 10:59 PM | #1 |
|
|
finding upper and lower sums of a region
here is my problem: find the upper and lower sums for the region bounded by the graph of [tex]f(x) = x^2[/tex] and the x-axis between x=0 and x=2. I understand what this problem is asking but i don't understand how to compte the left and right endpoints. the left endpoint is the following:
[tex]m_{i}=0+(i-1)\frac{2}_{n}[/tex] = [tex]\frac{2(i-1)}_{n}[/tex] and the right enpoint is given as this: [tex]M_{i}=0+i\frac{2}_{n}[/tex] = [tex]\frac{2i}_{n}[/tex] The lower sum ( left enpoint) is the following: 8\3 - 4\n + 4\3n^2 and the right endpoint is computed in a similar fashion. my question is, how do i find the left and right enpoints? what is the formula for doing this? what if the lower bound is not zero but some other number, how would i find it then? |
| Dec7-04, 12:25 AM | #2 |
|
Recognitions:
|
The lower (and upper) endpoints depend on n. For a given n, the [itex]m_i[/itex] you gave gives all the left endpoints if you let i range from 1 to n (same for the upper sum). Maybe working out a couple examples wouldn't hurt, if n=2 or 3 can you find the endpoints?
What are you planning on using for the lower sum? In your formulas for the left and right endpoints, the 0 is the leftmost point in the interval, the "2/n" is signifying the width of each sub-interval. If you are splitting the interval [a,b] up into n pieces, what would you replace these with? |
| Dec7-04, 07:38 AM | #3 |
|
|
I don't understand your question: you say, correctly, that
"the left endpoint is the following: = [tex]\frac{2(i-1)}_{n}[/tex]" and "the right endpoint is given as this [tex]\frac{2i}_{n}[/tex]" So what could you possibly mean by "how do i find the left and right enpoints?" Didn't you just do that? |
| Dec7-04, 09:59 AM | #4 |
|
|
finding upper and lower sums of a region
this problem solution was copied from my text. i did not fing the endpoints myself.
|
| Dec7-04, 12:52 PM | #5 |
|
|
You are asked to find the integral from 0 to 2 and are dividing that into n intervals.
The length of each interval, therefore, is the distance from 0 to 2 (which is 2) divided by n: 2/n. For example, if you only used 2 intervals, each would have length 1, of course. And your "endpoints" would be 0, 1, 2: start from the left at 0 and add that length. If you used 4 intervals, each would have length 2/4= 1/2 and your endpoints would be 0+ 0*(1/2)= 0, 0+ 1*(1/2)= 1/2, 0+ 2*(1/2)= 1, 0+ 3*(1/2)= 3/2, and 0+ 4*(1/2)= 2. If you use n intervals, each would have length 2/n and your endpoints would be 0+ 0*(2/n)=0, 0+ 1*(2/n)= 2/n, ..., 0+ i*(1/n)= 2/n. To get the right endpoints start with i= 1 and go up to n. To get the left endpoints, you could just use i= 0 to n-1. What your text is doing apparently, is replacing i in 2i/n by i-1 which gives 2(i-1)/n and you can use i= 1 to n again. |
| Dec8-04, 08:16 AM | #6 |
|
|
thanks IVY, I still don't quite understand what's going on with this idea,but i really appreciate the feedback.
|
| Thread Closed |
| Thread Tools | |
Similar Threads for: finding upper and lower sums of a region
|
||||
| Thread | Forum | Replies | ||
| Finding upper and lower limits of a graph | Precalculus Mathematics Homework | 2 | ||
| Please help!!! Struggling with finding the area using upper & lower sums!! | Calculus & Beyond Homework | 4 | ||
| Finding upper and lower sums of a region...Please HELP!!! | Calculus & Beyond Homework | 0 | ||
| upper and lower sums | Calculus & Beyond Homework | 7 | ||
| lower and upper sums | Calculus | 7 | ||