Main content
AP®︎/College Calculus AB
Course: 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
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice
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 between and .
And say we decide to do that by writing the expression for a right Riemann sum with four equal subdivisions, using summation notation.
Let denote the area of the rectangle in our approximation.
The entire Riemann sum can be written as follows:
What we need to do now is find the expression for .
The width of the entire interval is units and we want equal subdivisions, so the of each rectangle is units.
The of each rectangle is the value of at the right endpoint of the rectangle (because this is a right Riemann sum).
Let denote the right endpoint of the rectangle. To find for any value of , we start at (the left endpoint of the interval) and add the common width repeatedly.
Therefore, the formula of is . Now, the of each rectangle is the value of at its right endpoint:
And so we've arrived at a general expression for the area of the rectangle:
Now all we have left is to sum this expression for values of from to :
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 over the interval with equal subdivisions.
Define : Let denote the of each rectangle, then .
Define : Let denote the right endpoint of each rectangle, then .
Define area of rectangle: The of each rectangle is then , and the area of each rectangle is .
Sum the rectangles: Now we use summation notation to add all the areas. The values we use for are different for left and right Riemann sums:
- When we are writing a right Riemann sum, we will take values of
from to . - However, when we are writing a left Riemann sum, we will take values of
from to (these will give us the value of at the 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?(8 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.(4 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?(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".(3 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)
- 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)
- 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)
- 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(2 votes) - Is there a way to write summation notation for upper sums and lower sums?(2 votes)