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### Course: AP®︎/College Calculus BC > Unit 6

Lesson 3: Riemann sums, summation notation, and definite integral notation- Summation notation
- Summation notation
- Worked examples: Summation notation
- Summation notation
- Riemann sums in summation notation
- Riemann sums in summation notation
- Worked example: Riemann sums in summation notation
- Riemann sums in summation notation
- Definite integral as the limit of a Riemann sum
- Definite integral as the limit of a Riemann sum
- Worked example: Rewriting definite integral as limit of Riemann sum
- Worked example: Rewriting limit of Riemann sum as definite integral
- Definite integral as the limit of a Riemann sum

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# Riemann sums in summation notation

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 ${i}^{\text{th}}$ rectangle in our approximation.

The entire Riemann sum can be written as follows:

What we need to do now is find the expression for $A(i)$ .

The width of the entire interval $[0.5,3.5]$ is $3$ units and we want $4$ equal subdivisions, so the ${\text{width}}$ of each rectangle is $3\xf74={0.75}$ units.

The ${\text{height}}$ 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 ${i}^{\text{th}}$ 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 ${{x}_{i}}$ is ${0.5+0.75i}$ . Now, the ${\text{height}}$ of each rectangle is the value of $f$ at its right endpoint:

And so we've arrived at a general expression for the area of the ${i}^{\text{th}}$ rectangle:

Now all we have left is to sum this expression for values of $i$ from $1$ to $4$ :

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**$\mathrm{\Delta}x$ :Let

**Define**${x}_{i}$ :Let

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

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

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

Left Riemann sum | Right Riemann sum |
---|---|

*Want more practice? Try this exercise.*

## Want to join the conversation?

- What about midpoint sums? What is that notation?(9 votes)
- 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?(5 votes)
- 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.(5 votes)

- 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(5 votes)- 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.(4 votes)

- Question 1 says "what is the length of each rectangle, delta x?" Shouldn't it say what is the WIDTH?(6 votes)
- 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.(1 vote)

- 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?(5 votes)
- 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".(2 votes)

- 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?(5 votes)
- Reimann sums are a simple approximation of the area, not a concrete estimation. However, the anti-derivative is better, because it gives concrete answers.(1 vote)

- Is there a formula to find the riemann sum formula for a midpoint or trapezoidal r. sum? Thanks!(4 votes)
- Fifteen and three quarters(0 votes)

- 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 ?(3 votes)
- Is there a way to write summation notation for upper sums and lower sums?(2 votes)
- In the 1 problem (also in the review section) I am wondering if what they ask for is rather the
**left**endpoint, not the right.

https://imgur.com/iGj6uHF

The answer is:`x_i = 2 + 0.5 i`

When i=0, the result is 2, which is the left endpoint of the first rectangle.

Or, do I misunderstand something?(2 votes)