Key Concept

#Riemann Sums Study Notes

#Table of Contents

Introduction to Riemann Sums
Types of Riemann Sums
Worked Examples
Practice Questions
Glossary
Summary and Key Takeaways
Exam Strategy

Key Concept

#Introduction to Riemann Sums

A Riemann sum is a technique for approximating the exact value of an accumulation of change, which is often represented as a definite integral or the area between a curve and the x-axis. The basic idea is to sum up the areas of multiple rectangles under a curve to estimate this value.

Key Concept

#Types of Riemann Sums

There are three main types of Riemann sums:

Left Riemann Sum
Right Riemann Sum
Midpoint Riemann Sum

#Left Riemann Sum

#How to Calculate a Left Riemann Sum

To calculate the left Riemann sum of a function $f$ between $x=a$ and $x=b$ (where $a < b$ ):

Divide the interval into $n$ subintervals by choosing values $x_0, x_1, ..., x_n$ such that $a = x_0 < x_1 < \cdots < x_n = b$ .
Let this define $n$ rectangles.
The width of the $i$ -th rectangle is $(x_i - x_{i-1})$ , and the height is $f(x_{i-1})$ .
The left Riemann sum is the sum of the areas of these $n$ rectangles:

$\sum_{i=1}^{n} (x_i - x_{i-1}) \cdot f(x_{i-1})$

**Example:**

A social sciences researcher is using a function $m$ to model the total mass of garden gnomes appearing on lawns in a neighborhood over time $t$ . The table below gives selected values of $m'(t)$ , the rate of change of the mass, over the time interval $0 \le t \le 12$ . At time $t=0$ , $m(0) = 24.9$ kilograms.

t (days)	0	3	7	10	12
$m'(t)$ (kg/day)	2.6	4.8	12.2	0.7	-1.3

To find the left Riemann sum:

 $(3-0) \cdot 2.6 + (7-3) \cdot 4.8 + (10-7) \cdot ...$

Previous Topic - Accumulation of Change Next Topic - Trapezoidal Sums

Riemann Sums & Definite Integrals

David Brown

#Riemann Sums Study Notes

#Table of Contents

#Introduction to Riemann Sums

#Types of Riemann Sums

#Left Riemann Sum

#How to Calculate a Left Riemann Sum

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