Please help with the questions on Riemann sum

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Haru

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Hello, I am Haru and I have difficulties to solve the following problems. It would be much appreciated if someone can help me. Sorry for asking so many questions. And wish you Merry Christmas.

1. (a.) On a sketch of y=e^(x), represent the left Riemann sum with n=2 approximating ∫e^(x)dx (which the lower bound (from): 2 and the upper bound (to): 3). Write out the
terms of the sum, but do not evaluate it: Sum= + ?
(b.) On another sketch, represent the right Riemann sum with n=2n=2 approximating ∫e^(x)dx (which the lower bound (from): 2 and the upper bound (to): 3). Write out
the terms of the sum, but do not evaluate it: Sum = + ?
~ For these question, I do not even know where to start.
 
Haru said:
Hello, I am Haru and I have difficulties to solve the following problems. It would be much appreciated if someone can help me. Sorry for asking so many questions. And wish you Merry Christmas.

1. (a.) On a sketch of y=e^(x), represent the left Riemann sum with n=2 approximating ∫e^(x)dx (which the lower bound (from): 2 and the upper bound (to): 3). Write out the
terms of the sum, but do not evaluate it: Sum= + ?
(b.) On another sketch, represent the right Riemann sum with n=2n=2 approximating ∫e^(x)dx (which the lower bound (from): 2 and the upper bound (to): 3). Write out
the terms of the sum, but do not evaluate it: Sum = + ?
~ For these question, I do not even know where to start.
Click to expand...
And Happy Holidays to you!

Since you do not know where to start - let's start with definitions. For Riemanian sum, go to:

https://en.wikipedia.org › wiki › Riemann_sum


Please read the article and try the problems again. Please let us know if you do not understand some particular part of it.
 
Thank you. Now I have some idea for the Riemann Sum, however, I do not understand the following:

1. The following formula is found in the section called : Right Riemann Sum
{\displaystyle \left\vert \int _{a}^{b}f(x)\,dx-A_{\mathrm {right} }\right\vert \leq {\frac {M_{1}(b-a)^{2}}{2n}}}


~ For this formula, I do not understand what does the symbol : Aright means, does it mean the area covered by the right end-point?

2. For the "Example" part, I do not understand why we have to square Xi as shown in the following:
Taking an example, the area under the curve of y = x2 between 0 and 2 can be procedurally computed using Riemann's method.
The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of
{\tfrac {2}{n}}
; these are the widths of the Riemann rectangles (hereafter "boxes"). Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes will be
x_{1},x_{2},\ldots ,x_{n}
. Therefore, the sequence of the heights of the boxes will be
x_{1}^{2},x_{2}^{2},\ldots ,x_{n}^{2}
. It is an important fact that
x_{i}={\tfrac {2i}{n}}
, and
x_{n}=2
.
The area of each box will be
{\tfrac {2}{n}}\times x_{i}^{2}
and therefore the nth right Riemann sum will be:

{\begin{aligned}S&={\frac {2}{n}}\times \left({\frac {2}{n}}\right)^{2}+\cdots +{\frac {2}{n}}\times \left({\frac {2i}{n}}\right)^{2}+\cdots +{\frac {2}{n}}\times \left({\frac {2n}{n}}\right)^{2}\\&={\frac {8}{n^{3}}}\left(1+\cdots +i^{2}+\cdots +n^{2}\right)\\&={\frac {8}{n^{3}}}\left({\frac {n(n+1)(2n+1)}{6}}\right)\\&={\frac {8}{n^{3}}}\left({\frac {2n^{3}+3n^{2}+n}{6}}\right)\\&={\frac {8}{3}}+{\frac {4}{n}}+{\frac {4}{3n^{2}}}\end{aligned}}


~ I do not understand why we have to square Xi as the boxes should be rectangles which the area for rectangle is width x height=
{\tfrac {2}{n}}
x square of Xi but actually it is like the above and this make me have confusion.

Besides, after reading this article, I have some idea for the questions that I have posted yesterday now, which I would like to tell here so that I can know whether it is correct or not. And it would be much appreciated if we could discuss on them if they are wrong. Thank you.

First, for 1(a.) as the width for each box= (3-2)/2 = 1/2, but I still can't figure out the height of the boxes and I only know approximating the function by its value at the left-end point (which is 2 in this case) gives multiple rectangles. Also, I do not know how many area of rectangular boxes I have to add even though I have plotted a graph for y=e^(x).

Second, I think similar way can be applied into 1 (b.) to find the Right Riemann Sum.

And that are what I am thinking regarding to the article and the questions. Thank you.
 
Haru said:
Thank you. Now I have some idea for the Riemann Sum, however, I do not understand the following:

1. The following formula is found in the section called : Right Riemann Sum
{\displaystyle \left\vert \int _{a}^{b}f(x)\,dx-A_{\mathrm {right} }\right\vert \leq {\frac {M_{1}(b-a)^{2}}{2n}}}


~ For this formula, I do not understand what does the symbol : Aright means, does it mean the area covered by the right end-point?

2. For the "Example" part, I do not understand why we have to square Xi as shown in the following:
Taking an example, the area under the curve of y = x2 between 0 and 2 can be procedurally computed using Riemann's method.
The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of
{\tfrac {2}{n}}
; these are the widths of the Riemann rectangles (hereafter "boxes"). Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes will be
x_{1},x_{2},\ldots ,x_{n}
. Therefore, the sequence of the heights of the boxes will be
x_{1}^{2},x_{2}^{2},\ldots ,x_{n}^{2}
. It is an important fact that
x_{i}={\tfrac {2i}{n}}
, and
x_{n}=2
.
The area of each box will be
{\tfrac {2}{n}}\times x_{i}^{2}
and therefore the nth right Riemann sum will be:

{\begin{aligned}S&={\frac {2}{n}}\times \left({\frac {2}{n}}\right)^{2}+\cdots +{\frac {2}{n}}\times \left({\frac {2i}{n}}\right)^{2}+\cdots +{\frac {2}{n}}\times \left({\frac {2n}{n}}\right)^{2}\\&={\frac {8}{n^{3}}}\left(1+\cdots +i^{2}+\cdots +n^{2}\right)\\&={\frac {8}{n^{3}}}\left({\frac {n(n+1)(2n+1)}{6}}\right)\\&={\frac {8}{n^{3}}}\left({\frac {2n^{3}+3n^{2}+n}{6}}\right)\\&={\frac {8}{3}}+{\frac {4}{n}}+{\frac {4}{3n^{2}}}\end{aligned}}


~ I do not understand why we have to square Xi as the boxes should be rectangles which the area for rectangle is width x height=
{\tfrac {2}{n}}
x square of Xi but actually it is like the above and this make me have confusion.

Besides, after reading this article, I have some idea for the questions that I have posted yesterday now, which I would like to tell here so that I can know whether it is correct or not. And it would be much appreciated if we could discuss on them if they are wrong. Thank you.

First, for 1(a.) as the width for each box= (3-2)/2 = 1/2, but I still can't figure out the height of the boxes and I only know approximating the function by its value at the left-end point (which is 2 in this case) gives multiple rectangles. Also, I do not know how many area of rectangular boxes I have to add even though I have plotted a graph for y=e^(x).

Second, I think similar way can be applied into 1 (b.) to find the Right Riemann Sum.

And that are what I am thinking regarding to the article and the questions. Thank you.
Click to expand...
You said:

~ For this formula, I do not understand what does the symbol : Aright means, does it mean the area covered by the right end-point?

It means - the area of the rectangle considering the height of the rectangle [f(x)] at the right end-point of the small division (xi) you are considering.

Please, get this concept totally clarified in your mind - because you'll use this many times. You may want to sit down with your instructor and discuss this concept.
 
Haru said:
Thank you. Now I have some idea for the Riemann Sum, however, I do not understand the following:

1. The following formula is found in the section called : Right Riemann Sum
{\displaystyle \left\vert \int _{a}^{b}f(x)\,dx-A_{\mathrm {right} }\right\vert \leq {\frac {M_{1}(b-a)^{2}}{2n}}}


~ For this formula, I do not understand what does the symbol : Aright means, does it mean the area covered by the right end-point?
Click to expand...
I too was a bit confused initially as what Arightmeant. Then I thought about what is going on. The error they speak of must be between the actual area (which is the integral) and the estimated area (which is Aright). So what else can be meant by Aright? It could be the left endpoint rule of the midpoint rule but why put Aright for these cases. Surely it is the right hand rule being used.
 
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