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## AP®︎/College Calculus AB

### 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

AP.CALC:

LIM‑5 (EU)

, LIM‑5.B (LO)

, LIM‑5.B.2 (EK)

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, left parenthesis, x, right parenthesis, equals, square root of, x, end square root between x, equals, 0, point, 5 and x, equals, 3, point, 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, left parenthesis, i, right parenthesis denote the area of the i, start superscript, start text, t, h, end text, end superscript 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, left parenthesis, i, right parenthesis.

The width of the entire interval open bracket, 0, point, 5, comma, 3, point, 5, close bracket is 3 units and we want 4 equal subdivisions, so the start color #1fab54, start text, w, i, d, t, h, end text, end color #1fab54 of each rectangle is 3, divided by, 4, equals, start color #1fab54, 0, point, 75, end color #1fab54 units.

The start color #e07d10, start text, h, e, i, g, h, t, end text, end color #e07d10 of each rectangle is the value of f at the right endpoint of the rectangle (because this is a right Riemann sum).

Let start color #11accd, x, start subscript, i, end subscript, end color #11accd denote the right endpoint of the i, start superscript, start text, t, h, end text, end superscript rectangle. To find x, start subscript, i, end subscript for any value of i, we start at x, equals, 0, point, 5 (the left endpoint of the interval) and add the common width start color #1fab54, 0, point, 75, end color #1fab54 repeatedly.

Therefore, the formula of start color #11accd, x, start subscript, i, end subscript, end color #11accd is start color #11accd, 0, point, 5, plus, 0, point, 75, i, end color #11accd. Now, the start color #e07d10, start text, h, e, i, g, h, t, end text, end color #e07d10 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, start superscript, start text, t, h, end text, end superscript 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 open bracket, a, comma, b, close bracket with n equal subdivisions.

**Define delta, x:**Let start color #1fab54, delta, x, end color #1fab54 denote the start color #1fab54, start text, w, i, d, t, h, end text, end color #1fab54 of each rectangle, then start color #1fab54, delta, x, equals, start fraction, b, minus, a, divided by, n, end fraction, end color #1fab54.

**Define x, start subscript, i, end subscript:**Let start color #11accd, x, start subscript, i, end subscript, end color #11accd denote the right endpoint of each rectangle, then start color #11accd, x, start subscript, i, end subscript, equals, a, plus, delta, x, dot, i, end color #11accd.

**Define area of i, start superscript, start text, t, h, end text, end superscript rectangle:**The start color #e07d10, start text, h, e, i, g, h, t, end text, end color #e07d10 of each rectangle is then start color #e07d10, f, left parenthesis, start color #11accd, x, start subscript, i, end subscript, end color #11accd, right parenthesis, end color #e07d10, and the area of each rectangle is start color #1fab54, delta, x, end color #1fab54, dot, start color #e07d10, f, left parenthesis, start color #11accd, x, start subscript, i, end subscript, end color #11accd, right parenthesis, end color #e07d10.

**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, minus, 1 (these will give us the value of f at the*left*endpoint of each rectangle).

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

sum, start subscript, i, equals, 0, end subscript, start superscript, n, minus, 1, end superscript, start color #1fab54, delta, x, end color #1fab54, dot, start color #e07d10, f, left parenthesis, start color #11accd, x, start subscript, i, end subscript, end color #11accd, right parenthesis, end color #e07d10 | sum, start subscript, i, equals, 1, end subscript, start superscript, n, end superscript, start color #1fab54, delta, x, end color #1fab54, dot, start color #e07d10, f, left parenthesis, start color #11accd, x, start subscript, i, end subscript, end color #11accd, right parenthesis, end color #e07d10 |

*Want more practice? Try this exercise.*

## Want to join the conversation?

- 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)

- 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?(4 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)

- What about midpoint sums? What is that notation?(3 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?(3 votes)
- Is there a formula to find the riemann sum formula for a midpoint or trapezoidal r. sum? Thanks!(3 votes)
- Fifteen and three quarters(1 vote)

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