How to Integrate Upper Bounds with Circumscribed Rectangles

In summary, the homework statement asks for an integration from 2 to 5 using the limit with circumscribed rectangles. The attempt at a solution finds that A=lim(4/n)(4/n)(4)(2+3+4+...+(n+1)) which is 64/n^2((n^2+3n)/2))= 32lim((n+3)/n))=48. However, the answer is obviously 48 because the limit is 1 to 5, not 2 to 5.
  • #1
tmclary
13
0
[SOLVED] Upper Bounds Integration

Homework Statement


Integrate y=4x from 2 to 5 using the limit with circumscribed rectangles.


Homework Equations



A=lim(n to inf.) Summation of f(xsubi) times delta (xsubi)

The Attempt at a Solution



A=lim(4/n)(4/n)(4)(2+3+4+...+(n+1))
=64/n^2((n^2+3n)/2))= 32lim((n+3)/n)) =32. But from integration the answer is obviously 48. What am I doing wrong? (Sorry about lack of typo skills-newbie)
 
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  • #2
why do you think the answer is obviously 48?
Your answer isn't correct again thout..
 
  • #3
well i got 42 as my answer, either by directly integrating

[tex]\int_2^5 4xdx[/tex] and also by using Rieman sums.

I'll try to post my work, on my next post.
 
  • #4
Sorry-wrong limits!

Sorry! The limits were 1 to 5, not 2 to 5!
 
  • #5
we want to calculate

[tex]\lim_{n\to\infty}\sum_{i=1}^{n}f(\epsilon_i)\delta x_i[/tex]

now let us create n mini segments on the segment [2,5]

that is let the points be

[tex]x_0=2,x_1,x_2,...x_i_-_1,x_i,...,x_n=5[/tex]

Now our concern is to determine what our function will be.
First let's notice certian facts:

[tex]\delta x_i=x_i-x_i_-_1[/tex] also let [tex]\epsilon_i=x_i[/tex]

this way we have:

[tex]\epsilon_i=\delta x_i+x_i_-_1[/tex]

also: [tex]\delta x_i=\frac{5-2}{n}=\frac{3}{n}[/tex]

Now, for to determine our function let's try some values for i=1,2,3,...

[tex]f(x_1)=4\left(\frac{3}{n}+2\right),f(x_2)=4(\frac{6}{n}+2),f(x_3)=4(\frac{9}{n}+2),..., f(x_i)=4(\frac{3i}{n}+2)[/tex]


Hence:

[tex]\int_2^54xdx=\lim_{n\to\infty}\sum_{i=1}^{n}4\left(\frac{3i}{n}+2\right)\frac{3}{n}=...=42[/tex]
 
Last edited:
  • #6
tmclary said:
Sorry! The limits were 1 to 5, not 2 to 5!

Well, then do the same thing as i did here, just take into consideration that you have the lower limit 1, in this case. I am not going to troube to go the same route again, i think you can do it now. If you can't ask again.

cheers!
 
  • #7
Well it doesn't change a lot by the way, the difference is that now you'll have

[tex]\delta x_i=\frac{4}{n}[/tex] and

[tex]f(x_i)=4\left(\frac{4i}{n}+1\right)[/tex]

and the answer will be 48.
 
  • #8
Thanks very much for your replies- I'm still stuck expanding the summation- will attempt another query when I have time, and can clarify.
 
  • #9
Got it- I wasn't adding the 1 to the 4/n. Thanks again for your answer.
 
  • #10
tmclary said:
Got it- I wasn't adding the 1 to the 4/n. Thanks again for your answer.

I tried to post a detailed answer, including how the summation expanded and all that stuff, but after i typed it all, i don't know for some crappy reason it did not show up. Anyways, I'm glad you got it !
 

What is Upper Bounds Integration?

Upper bounds integration is a mathematical technique used to find the maximum or upper limit of a function or set of data. It involves finding the largest possible value that a function can reach within a given interval.

What is the purpose of using Upper Bounds Integration?

The purpose of using upper bounds integration is to determine the maximum value of a function or set of data. This can be useful in many applications, such as optimization problems, where finding the maximum value is important.

How is Upper Bounds Integration calculated?

Upper bounds integration is calculated by dividing the interval into smaller subintervals and finding the maximum value within each subinterval. Then, the maximum values from each subinterval are compared to find the overall maximum value within the entire interval.

What are some real-world applications of Upper Bounds Integration?

Some real-world applications of upper bounds integration include finding the maximum profit or revenue for a company, determining the maximum amount of weight a bridge can hold, and finding the maximum amount of energy that can be generated from a power plant.

Is Upper Bounds Integration the same as finding the absolute maximum?

No, upper bounds integration is not the same as finding the absolute maximum. Absolute maximum refers to the largest value of a function over its entire domain, while upper bounds integration only considers the maximum value within a specific interval.

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