I am reading from another book that has slightly different notation. It has f(x with a superscript* and subscript i). First I am not sure how to say it in English. Second is the asterisk on the x or the i? Would it be f of x star sub i, or f of x sub i star?

I'd actually just say "f of x sub i". I assume the notation's purpose is to denote a general height, rather than go into the details of a left-hand, right-hand, or midpoint sum, which "f of x sub i" accomplishes.

What is the length of each rectangle, \greenD{\Delta x}Δxstart color greenD, delta, x, end color greenD? how do you gett the answer so you can gett to the next question? What ever i whrigt its the wrong answer. So how can i learn if i dont get the right one? I love the videos by the way! life saver

The question asks for the length of each rectangle, which is the width of each subdivision. On an interval with endpoints a and b, where we need n subdivisions, the width of each subdivision is (b-a)/n. So for this problem, we have the interval [2,7] and we need 10 subdivisions. We find the width of each rectangle by doing (7-2)/10.

At problem 2 we want to approximate the area between g(x) and the x-axis. However in the solution (Explain) f(x_i) is calculated. Shouldn't it be g(x_i) instead?

It should be g(x_i). Good eye. You can report the mistake by clicking on the "Ask a question" box and selecting "Report a Mistake".

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AP®︎/College Calculus AB

Riemann sums in summation notation

Google Classroom

Summation notation can be used to write Riemann sums in a compact way. This is a challenging, yet important step towards a formal definition of the definite integral.

Summation notation (or sigma notation) allows us to write a long sum in a single expression. While summation notation has many uses throughout math (and specifically calculus), we want to focus on how we can use it to write Riemann sums.

Example of writing a Riemann sum in summation notation

Imagine we are approximating the area under the graph of

f (x) = \sqrt{x}

between

x = 0.5

and

x = 3.5

And say we decide to do that by writing the expression for a right Riemann sum with four equal subdivisions, using summation notation.

Let

A (i)

denote the area of the

i^{th}

rectangle in our approximation.

The entire Riemann sum can be written as follows:

A (1) + A (2) + A (3) + A (4) = \sum_{i = 1}^{4} A (i)

What we need to do now is find the expression for

A (i)

The width of the entire interval

[0.5, 3.5]

3

units and we want

4

equal subdivisions, so the

width

of each rectangle is

3 \div 4 = 0.75

units.

The

height

of each rectangle is the value of

f

at the right endpoint of the rectangle (because this is a right Riemann sum).

Let

x_{i}

denote the right endpoint of the

i^{th}

rectangle. To find

x_{i}

for any value of

i

, we start at

x = 0.5

(the left endpoint of the interval) and add the common width

0.75

repeatedly.

Therefore, the formula of

x_{i}

0.5 + 0.75 i

. Now, the

height

of each rectangle is the value of

f

at its right endpoint:

f (x_{i}) = \sqrt{x_{i}} = \sqrt{0.5 + 0.75 i}

And so we've arrived at a general expression for the area of the

i^{th}

rectangle:

\begin{aligned} A (i) & = width \cdot height \\ = 0.75 \cdot \sqrt{0.5 + 0.75 i} \end{aligned}

Now all we have left is to sum this expression for values of

i

from

1

4

\begin{aligned} A (1) + A (2) + A (3) + A (4) \\ = \sum_{i = 1}^{4} A (i) \\ = \sum_{i = 1}^{4} 0.75 \cdot \sqrt{0.5 + 0.75 i} \end{aligned}

And we're done!

Summarizing the process of writing a Riemann sum in summation notation

Imagine we want to approximate the area under the graph of

f

over the interval

[a, b]

with

n

equal subdivisions.

Define $Δ x$ ‍: Let

Δ x

denote the

width

of each rectangle, then

Δ x = \frac{b - a}{n}

Define $x_{i}$ ‍: Let

x_{i}

denote the right endpoint of each rectangle, then

x_{i} = a + Δ x \cdot i

Define area of $i^{th}$ ‍ rectangle: The

height

of each rectangle is then

f (x_{i})

, and the area of each rectangle is

Δ x \cdot f (x_{i})

Sum the rectangles: Now we use summation notation to add all the areas. The values we use for

i

are different for left and right Riemann sums:

When we are writing a right Riemann sum, we will take values of $i$ ‍ from $1$ ‍ to $n$ ‍.
However, when we are writing a left Riemann sum, we will take values of $i$ ‍ from $0$ ‍ to $n - 1$ ‍ (these will give us the value of $f$ ‍ at the left endpoint of each rectangle).

Left Riemann sum	Right Riemann sum
$\sum_{i = 0}^{n - 1} Δ x \cdot f (x_{i})$ ‍	$\sum_{i = 1}^{n} Δ x \cdot f (x_{i})$ ‍

Problem 1.A

Problem set 1 will walk you through the process of approximating the area between

f (x) = 0.1 x^{2} + 1

and the

x

-axis on the interval

[2, 7]

using a left Riemann sum with

10

equal subdivisions.

What is the length of each rectangle, $Δ x$ ‍?

Δ x =

Problem 2

We want to approximate the area between

g (x) = \frac{5}{x} + 2

and the

x

-axis on the interval

[1, 7]

using a right Riemann sum with

9

equal subdivisions:

Which expression represents our approximation?

Want more practice? Try this exercise.

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gingerseal8
Posted 2 years ago. Direct link to gingerseal8's post “What about midpoint sums?...”
What about midpoint sums? What is that notation?
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(8 votes)
Answer
john.coder
Posted 5 years ago. Direct link to john.coder's post “I am reading from another...”
I am reading from another book that has slightly different notation. It has f(x with a superscript* and subscript i). First I am not sure how to say it in English. Second is the asterisk on the x or the i? Would it be f of x star sub i, or f of x sub i star?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Alex
  Posted 5 years ago. Direct link to Alex's post “I'd actually just say "f ...”
  I'd actually just say "f of x sub i". I assume the notation's purpose is to denote a general height, rather than go into the details of a left-hand, right-hand, or midpoint sum, which "f of x sub i" accomplishes.
  Button navigates to signup page
  (4 votes)
Lina Johansson
Posted 5 years ago. Direct link to Lina Johansson's post “What is the length of eac...”
What is the length of each rectangle, \greenD{\Delta x}Δxstart color greenD, delta, x, end color greenD?

how do you gett the answer so you can gett to the next question?
What ever i whrigt its the wrong answer.
So how can i learn if i dont get the right one?

I love the videos by the way! life saver
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Ardaschir
  Posted 5 years ago. Direct link to Ardaschir's post “The question asks for the...”
  The question asks for the length of each rectangle, which is the width of each subdivision. On an interval with endpoints a and b, where we need n subdivisions, the width of each subdivision is (b-a)/n. So for this problem, we have the interval [2,7] and we need 10 subdivisions. We find the width of each rectangle by doing (7-2)/10.
  Comment on Ardaschir's post “The question asks for the...”
  (4 votes)
kosakkas
Posted 5 years ago. Direct link to kosakkas's post “At problem 2 we want to a...”
At problem 2 we want to approximate the area between g(x) and the x-axis. However in the solution (Explain) f(x_i) is calculated. Shouldn't it be g(x_i) instead?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Alex
  Posted 5 years ago. Direct link to Alex's post “It should be g(x_i). Good...”
  It should be g(x_i). Good eye. You can report the mistake by clicking on the "Ask a question" box and selecting "Report a Mistake".
  Button navigates to signup page
  (3 votes)
Krispy
Posted 2 years ago. Direct link to Krispy's post “Question 1 says "what is ...”
Question 1 says "what is the length of each rectangle, delta x?" Shouldn't it say what is the WIDTH?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Edwin Kim
  Posted 5 months ago. Direct link to Edwin Kim's post “Delta x is the width. Spe...”
  Delta x is the width. Specifically we sometimes refer to delta x as an "increment in x", therefore being the width of each of the rectangles.
  Button navigates to signup page
  (1 vote)
Steve
Posted 5 years ago. Direct link to Steve's post “So wait...I'm seeing conf...”
So wait...I'm seeing conflicting information... I see sometimes that using Reimann sums can work for Integrals. but that it's a longer and more complex way of doing Integrals(Using the Anti-deriative is better) But then I see that Riemann Sums can only give you an approximation or estimate of integrals? So which is it?
Button navigates to signup pageComment on Steve's post “So wait...I'm seeing conf...”
(5 votes)
Answer
Marquez, Ricardo
Posted 2 years ago. Direct link to Marquez, Ricardo's post “Is there a formula to fin...”
Is there a formula to find the riemann sum formula for a midpoint or trapezoidal r. sum? Thanks!
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(4 votes)
Answer
- Jason DiMartino
  Posted 2 years ago. Direct link to Jason DiMartino's post “Fifteen and three quarter...”
  Fifteen and three quarters
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  (0 votes)
James Kierstead
Posted 3 months ago. Direct link to James Kierstead's post “I don't understand how th...”
I don't understand how they got 15 over the line in the explanation for the last one. I.e. why did they go from 5/xi to 15/3+2i ?
Button navigates to signup pageComment on James Kierstead's post “I don't understand how th...”
(3 votes)
Answer
maximquantum
Posted 4 years ago. Direct link to maximquantum's post “Before the reveal of the ...”
Before the reveal of the formula:

n
∑ Δx⋅f(x<i>)
i=1

I made up a different formula which looks very much like the one above, but is it then wrong or is mine also legit?
the mine:

n
∑ f(i) * (b-a)/n
i=a
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(2 votes)
Answer
Timothy Nguyen
Posted 4 years ago. Direct link to Timothy Nguyen's post “Is there a way to write s...”
Is there a way to write summation notation for upper sums and lower sums?
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(2 votes)
Answer

AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 6

Example of writing a Riemann sum in summation notation

Summarizing the process of writing a Riemann sum in summation notation

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