Right endpoint approximation part II

Hckyplayer8

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Find a formula for f(x)=x2 +1 on the interval [0,1]. Then compute the area under the graph by evaluating the limit of Rn as N approaches infinity.

So we have the graph of a function and it's area under the graph. We want to partition up the area under the curve into infinitely smaller rectangles.

What does it mean, find the formula?

In my first post, the problem was similar, but I didn't need to find a formula to solve the question.
 

Dr.Peterson

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The question makes no sense to me. The formula for f(x) is given, namely x2 + 1. Are you sure you copied it exactly?

If so, I'd look in your book for an example that uses the word "formula" in a similar way. Maybe they mean to write a "formula" for the summation, which you are evidently calling Rn. If nothing else, you may need to show us an example from the book so we can see their terminology.

By the way, n and N are different things in mathematics; be careful!
 

Jomo

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You do want to partition the interval [0,1] into infinitely many equal partitions. NO! You want to partition [0,1] into N equal partitions. Do as you did before except there are N partitions. Then take the limit as n goes to infinity.
 

Jomo

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Find a formula for f(x)=x2 +1 on the interval [0,1]. Then compute the area under the graph by evaluating the limit of Rn as N approaches infinity.

So we have the graph of a function and it's area under the graph. We want to partition up the area under the curve into infinitely smaller rectangles.

What does it mean, find the formula?

In my first post, the problem was similar, but I didn't need to find a formula to solve the question.
Let the base be partitioned into N equal intervals. So the length of each base will be (1-0)/N = 1/N.
Now we want the right hand heights of each rectangle, They will be f(0 + 1/N) + f(0+2/N) + f(3/N) +...+f(N/N=1)
So we get the approximation for the area under the curve as (1/N)(f(0 + 1/N) + f(0+2/N) + f(3/N) +...+f(N/N=1))
=(1/N)[(1^2/N^2+1) + (2^2/N^2+1) + ... +(N^2/N^2 +1) ]= (1/N^3)[(1^2+2^2+...+N^2)] + (1+1+1+...+1)/N= (1/N^3)(N)(N+1)(2N+1)/6 +N/N = (N)(N+1)(2N+1)/(6N^3) + 1

Now calculate \(\displaystyle \lim_{N\to\infty} (\frac{(N)(N+1)(2N+1)}{6N^3} + 1)\)


BTW, I suspect what you are calling the formula is \(\displaystyle (\frac{(N)(N+1)(2N+1)}{6N^3} + 1)\)
 
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Hckyplayer8

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The question makes no sense to me. The formula for f(x) is given, namely x2 + 1. Are you sure you copied it exactly?

If so, I'd look in your book for an example that uses the word "formula" in a similar way. Maybe they mean to write a "formula" for the summation, which you are evidently calling Rn. If nothing else, you may need to show us an example from the book so we can see their terminology.

By the way, n and N are different things in mathematics; be careful!
Thanks for the reply. Here is the question, verbatim.

1.PNG
 

Jomo

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Hckyplayer8

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Let the base be partitioned into N equal intervals. So the length of each base will be (1-0)/N = 1/N.
Now we want the right hand heights of each rectangle, They will be f(0 + 1/N) + f(0+2/N) + f(3/N) +...+f(N/N=1)
So we get the approximation for the area under the curve as (1/N)(f(0 + 1/N) + f(0+2/N) + f(3/N) +...+f(N/N=1))
=(1/N)[(1^2/N^2+1) + (2^2/N^2+1) + ... +(N^2/N^2 +1) ]= (1/N^3)[(1^2+2^2+...+N^2)] + (1+1+1+...+1)/N= (1/N^3)(N)(N+1)(2N+1)/6 +N/N = (N)(N+1)(2N+1)/(6N^3) + 1

Now calculate \(\displaystyle \lim_{N\to\infty} (\frac{(N)(N+1)(2N+1)}{6N^3} + 1)\)


BTW, I suspect what you are calling the formula is \(\displaystyle (\frac{(N)(N+1)(2N+1)}{6N^3} + 1)\)
Hello Jomo.

So we break the x interval into equal but unspecified amount of partitions. To do so we find the difference of the x coords involved in the interval and divide by how many partitions we want. Since it is unspecified it ends up being 1/N.

Here is where my understanding breaks down. How am I suppose to find the right hand heights of the rectangles when they don't specify how many rectangles they want?
 

lev888

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Hello Jomo.

So we break the x interval into equal but unspecified amount of partitions. To do so we find the difference of the x coords involved in the interval and divide by how many partitions we want. Since it is unspecified it ends up being 1/N.

Here is where my understanding breaks down. How am I suppose to find the right hand heights of the rectangles when they don't specify how many rectangles they want?
It is specified. N rectangles. Did you read post #4?
 

Dr.Peterson

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When you work with a variable, you think of it as a specific number. You just don't know what it is, so you use its name.

So the number of subintervals is not "unspecified" while you do this work! It is the number N.

You have found that the width of each rectangle is the number 1/N. The first starts at 0, so its right-hand end is 0 + 1/N , that is, x = 1/N. Find the height of the curve there -- not, of course, as a specific number, but as an expression.

Do the same for each of the N intervals (in principle, of course, since you can't list all N of them).

I assume that, if you were given this assignment, you have also seen an example of this kind of problem, at least in part. Look at that to see how it is done.
 

Jomo

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As mentioned you are given the number of rectangles to use! It is N rectangles. I hope that you are not thinking that N is infinity as it is certainly not. N is some finite integer. As doing these problems think of N as 25, just do not write 25 but rather write N. For example each base will be \(\displaystyle \frac{1-0}{25}\) but write \(\displaystyle \frac{1-0}{N}\)

I can understand how this can confuse you but doesn't my example clear everything. Please go over it carefully and the instant that something is not clear, please do two things--First think about it and if it is still unclear ask about that specific step. We can help you but giving us exactly where you are getting stuck will make it much easier for us to help
 

Jomo

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Hckyplayer8,
So are you understanding the problem now? Let us know!
 

Hckyplayer8

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I appreciate all the help that has been provided to me.

I understand that N stands for some undesignated, positive integer. I understand that it does not stand for infinity. I even understand the concept of how we got the base of the rectangles to become 1/N by dividing the original interval [0,1] by N. But after that I feel like I'm oblivious to something that should be obvious.

I will "think aloud" and perhaps someone can rephrase something that will trigger my understanding.

The purpose of this problem can be broken down into two sections.

1. Find the formula for RN which is the sum of the squares of N.
2. Compute the area of the graph by solving for the limit of those squares.

Shouldn't there be some addition information?

When I am viewing this I am running a pseudo conversation in my head.

person 1: I need you to find the area under the curve of this function over this interval.
person 2: Cool I can do that. I just need to know how many rectangles I am breaking that interval up into.
person 1: Yes
person 2: Yes? That doesn't answer how many rectangles you want.
person 1: Yes

I've reviewed the section in my book and I understand the concept that as N increases, two things happen. The number of rectangles increase and that the approximation of the area becomes more correct.

How can we add up our sub-intervals if we never give them a definitive endpoint to begins with? N could be anything after all. Its a variable and the problem didn't specify how many rectangles it wants.
 

Jomo

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You said that each base, which goes from 0 to 1, has a length of 1/N. This is correct

Now starting from the left (ie from 0) tell us each interval for each base.

For example, if you wanted N=4 (ie 4 rectangles), then each base will be 1/4 unit in length. The 4 bases will (0,1/4),(1/4,2/4=1/2)(1/2,3/4) and finally (3/4, 4/4=1).

Again, please do what I should above for N

We need to do this in steps to assure that you understand it.
 

Hckyplayer8

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You said that each base, which goes from 0 to 1, has a length of 1/N. This is correct

Now starting from the left (ie from 0) tell us each interval for each base.

For example, if you wanted N=4 (ie 4 rectangles), then each base will be 1/4 unit in length. The 4 bases will (0,1/4),(1/4,2/4=1/2)(1/2,3/4) and finally (3/4, 4/4=1).

Again, please do what I should above for N

We need to do this in steps to assure that you understand it.
"Now starting from the left (ie from 0) tell us each interval for each base."

That is where my understanding breaks down. How can I add something that is not definitive? You want 1,2,3,4... rectangles, I can do that. Take the original interval and divide by how many rectangles you want will give you the bases.

But how can you find your sub intervals when the problem doesn't even specify how many rectangles it is aiming to achieve? Am I suppose to choose? I want 2 rectangles for a very poor approximation. I want 50 rectangles for a better approximation.

What is the point of this?
 

lev888

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"Now starting from the left (ie from 0) tell us each interval for each base."

That is where my understanding breaks down. How can I add something that is not definitive? You want 1,2,3,4... rectangles, I can do that. Take the original interval and divide by how many rectangles you want will give you the bases.

But how can you find your sub intervals when the problem doesn't even specify how many rectangles it is aiming to achieve? Am I suppose to choose? I want 2 rectangles for a very poor approximation. I want 50 rectangles for a better approximation.

What is the point of this?
What's the area of a rectangle with side lengths A and B?
 

Hckyplayer8

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