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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)=x between x=0.5 and x=3.5.
Function y = the square root of x is graphed. The x-axis goes from 0 to 4. The graph is a curve. The curve starts at (0, 0), moves upward concave down, and ends at (4, 2). The region between the curve and the x-axis, between x = 0.5 and x = 3.5, is shaded.
And say we decide to do that by writing the expression for a right Riemann sum with four equal subdivisions, using summation notation.
The graph of function y has the shaded region divided into 4 rectangles of width 0.75. Each rectangle touches the curve at the top right corner.
Let A(i) denote the area of the ith rectangle in our approximation.
The area of the rectangles are A of 1, A of 2, A of 3, and A of 4.
The entire Riemann sum can be written as follows:
A(1)+A(2)+A(3)+A(4)=i=14A(i)
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 width of each rectangle is 3÷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 xi denote the right endpoint of the ith rectangle. To find xi 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.
The left side of the first rectangle is at x = 0.5. Add 0.75 4 times to get the sides of the rectangles, at x sub 1 to x sub 4.
Therefore, the formula of xi is 0.5+0.75i. Now, the height of each rectangle is the value of f at its right endpoint:
f(xi)=xi=0.5+0.75i
And so we've arrived at a general expression for the area of the ith rectangle:
A(i)=widthheight=0.750.5+0.75i
Now all we have left is to sum this expression for values of i from 1 to 4:
=A(1)+A(2)+A(3)+A(4)=i=14A(i)=i=140.750.5+0.75i
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 Δx: Let Δx denote the width of each rectangle, then Δx=ban.
Define xi: Let xi denote the right endpoint of each rectangle, then xi=a+Δxi.
Define area of ith rectangle: The height of each rectangle is then f(xi), and the area of each rectangle is Δxf(xi).
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 n1 (these will give us the value of f at the left endpoint of each rectangle).
Left Riemann sumRight Riemann sum
i=0n1Δxf(xi)i=1nΔxf(xi)
Problem 1.A
Problem set 1 will walk you through the process of approximating the area between f(x)=0.1x2+1 and the x-axis on the interval [2,7] using a left Riemann sum with 10 equal subdivisions.
Function f is graphed. The x-axis goes from negative 1 to 9. The graph is a curve. The curve starts in quadrant 2, moves downward to a relative minimum at (0, 1), moves upward and ends in quadrant 1. The region between the curve and the x-axis, between x = 2 and x = 7, is shaded.
What is the length of each rectangle, Δx?
Δx=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Problem 2
We want to approximate the area between g(x)=5x+2 and the x-axis on the interval [1,7] using a right Riemann sum with 9 equal subdivisions:
Function g is graphed. The x-axis goes from negative 1 to 7. The graph is a curve. The curve starts in quadrant 1, moves downward concave up, and ends in quadrant 1. The region between the curve and the x-axis, between x = 1 and x = 7, is shaded. The shaded region is divided into 9 rectangles of equal width. Each rectangle touches the curve at the top right corner.
Which expression represents our approximation?
Choose 1 answer:

Want more practice? Try this exercise.

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