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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 $f\left(x\right)=\sqrt{x}$ 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 $A\left(i\right)$ denote the area of the ${i}^{\text{th}}$ rectangle in our approximation.
The entire Riemann sum can be written as follows:
$A\left(1\right)+A\left(2\right)+A\left(3\right)+A\left(4\right)=\sum _{i=1}^{4}A\left(i\right)$
What we need to do now is find the expression for $A\left(i\right)$.
The width of the entire interval $\left[0.5,3.5\right]$ is $3$ units and we want $4$ equal subdivisions, so the $\text{width}$ of each rectangle is $3÷4=0.75$ 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:
$f\left({x}_{i}\right)=\sqrt{{x}_{i}}=\sqrt{0.5+0.75i}$
And so we've arrived at a general expression for the area of the ${i}^{\text{th}}$ rectangle:
$\begin{array}{rl}A\left(i\right)& =\text{width}\cdot \text{height}\\ \\ & =0.75\cdot \sqrt{0.5+0.75i}\end{array}$
Now all we have left is to sum this expression for values of $i$ from $1$ to $4$:
$\begin{array}{rl}& \phantom{=}A\left(1\right)+A\left(2\right)+A\left(3\right)+A\left(4\right)\\ \\ & =\sum _{i=1}^{4}A\left(i\right)\\ \\ & =\sum _{i=1}^{4}0.75\cdot \sqrt{0.5+0.75i}\end{array}$
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 $\left[a,b\right]$ with $n$ equal subdivisions.
Define $\mathrm{\Delta }x$: Let $\mathrm{\Delta }x$ denote the $\text{width}$ of each rectangle, then $\mathrm{\Delta }x=\frac{b-a}{n}$.
Define ${x}_{i}$: Let ${x}_{i}$ denote the right endpoint of each rectangle, then ${x}_{i}=a+\mathrm{\Delta }x\cdot i$.
Define area of ${i}^{\text{th}}$ rectangle: The $\text{height}$ of each rectangle is then $f\left({x}_{i}\right)$, and the area of each rectangle is $\mathrm{\Delta }x\cdot f\left({x}_{i}\right)$.
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 sumRight Riemann sum
$\sum _{i=0}^{n-1}\mathrm{\Delta }x\cdot f\left({x}_{i}\right)$$\sum _{i=1}^{n}\mathrm{\Delta }x\cdot f\left({x}_{i}\right)$
Problem 1.A
Problem set 1 will walk you through the process of approximating the area between $f\left(x\right)=0.1{x}^{2}+1$ and the $x$-axis on the interval $\left[2,7\right]$ using a left Riemann sum with $10$ equal subdivisions.
What is the length of each rectangle, $\mathrm{\Delta }x$?
$\mathrm{\Delta }x=$

Problem 2
We want to approximate the area between $g\left(x\right)=\frac{5}{x}+2$ and the $x$-axis on the interval $\left[1,7\right]$ using a right Riemann sum with $9$ equal subdivisions:
Which expression represents our approximation?

Want more practice? Try this exercise.

## Want to join the conversation?

• What about midpoint sums? What is that notation?
• 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".
• Question 1 says "what is the length of each rectangle, delta x?" Shouldn't it say what is the WIDTH?
• 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?
• Is there a formula to find the riemann sum formula for a midpoint or trapezoidal r. sum? Thanks!
• Fifteen and three quarters
(1 vote)
• 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 ?
• 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