Understanding Riemann Sums (Your First Step In Integral Calculus)

Have you ever felt the desire to “UNDO” something?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching riemann sum

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

Consider how in math, every action has a REVERSE.

For example, subtraction is the “UNDO” for addition, and division is the REVERSE for multiplication.

Square roots “UNDO” squares, and logarithms REVERSE exponentials.

This concept continues in calculus with anti-differentiation reversing differentiation.

What’s that?

Well, it’s the antiderivative or, as I like to say, “the derivative in reverse” or “backwards derivatives.”

Isn’t it interesting how math consistently offers a way to backtrack? Much like that CTRL-Z shortcut on your keyboard (CMD-Z for you Mac Folks)

In other words, Anti-Differentiation is like CTRL-Z(ing) the derivative to return to the original function!

Your Key Concepts:

Antidifferentiation:

Antidifferentiation is reversing differentiation.
It means finding the original function when we know its rate of change. Put simply, if $f (x)$ shows how fast something changes.
Then its antiderivative $F (x)$ tells us the state (position) of that thing after some time, plus a constant $C$ since we might not know the starting point.

Riemann Sums:

Riemann sums are a way to guess the area under a curve by breaking it into simple shapes like rectangles or trapezoids.
The more shapes we use, the closer we reach the true area.
Different rules, like the Left Endpoint Rule or Midpoint Rule, guide us in estimating how big these shapes should be to mimic the curve’s shape.

Definite Integration:

Definite integration is about finding the precise area under a curve from point $a$ to point $b$ .
It refines the Riemann sums to the point where the rectangles (or other shapes) are infinitely thin, giving the exact result.
Mathematically, it calculates $F (b) - F (a)$ , the difference in the antiderivative $F (x)$ at the end and the start of the interval, which equals this exact area.

Now let’s learn more about each of these ideas…

Integration: The Smoother Way to Say “Anti differentiation”

But antidifferentiation and antiderivative are quite a mouthful to say, don’t you think?

Thankfully, another way of saying antidifferentiation is to say Indefinite Integration, or more simply, integration.

Before we go any further, we need to talk about two types of integration:

Indefinite
Definite

Indefinite Integration helps us to find the family of functions F(x) when we are given its derivative f(x).

$\int f (x) d x = F (x) + C$

Definite Integration is the process of finding a numerical value that represents the area under a curve over an interval [a,b].

Also known as the Riemann Integral.

$\int_{a}^{b} f (x) d x = F (b) - F (a)$

Next, we’ll explore how Riemann Integration works to give us these precise numbers.

Integration: The Tool for Area Measurement

One of the most useful things that integration does for us, besides undoing the derivative, is to calculate the area of a plane region!

In other words, it calculates the area under the curve!

Bernhard Riemann, a pioneering German mathematician, found an ingenious way to estimate the area under curves. He did this by breaking the area down into rectangles and tallying up their individual areas.

And then comes the ingenious twist!

If we divide the region into an infinite number of rectangles, such that the width of each rectangle is infinitesimally small, then the sum of these itty-bitty rectangular areas will equal the area of the plane region!

The approximation of the area under a curve is called a Riemann Sum.

Finding Areas and Distances

How Riemann Sums Help Us Measure Area

Let’s look at the notation you will utilize for Riemann Sums.

It’s straightforward, really: the area of a rectangle is just its width multiplied by its height.

So, if $Δ x$ represents the width of each rectangle in our approximation and $f (x)$ is the height, then multiplying these together gives us the area of one rectangle, $Δ x f (x)$ .

Thus, if we wish to estimate the area of a region, we will subdivide the region into a specified number of rectangles, $n$ , such that.

$\sum_{i = 0}^{n} (Δ x) (f (x_{i}))$

And this formula is called the Riemann Sum Approximation, which we will use to estimate the area under the curve.

✏️ Take note, though, there are a few ways to calculate this:

Right Endpoint
Left Endpoint
Midpoint Rule
Trapezoidal Rule

Technically, the Trapezoid Rule and its more advanced cousin, Simpson’s Rule, are not Riemann sum approximations.

But because their approximation and approach are strikingly similar to a Riemann Sum, we will learn how to wield their power along with the other estimations.

Riemann Sums: Calculating Areas Step by Step

How do we put Riemann Sums to practical use?

Imagine we have a curve defined by $f (x)$ within the interval [a,b].

Our goal is to subdivide the interval (region) into $n$ rectangles (or trapezoids if we’re using that approach) and then calculate the area using one of four methods:

Left
Right
Midpoint
Trapezoidal

Here is the straightforward, three-step process:

Determine the width ( $Δ x$ ) of each rectangle with the formula:

$Δ x = \frac{b - a}{n}$

Select a method to calculate the area, rectangle or trapezoid:

\begin{aligned} Area Rectangle = (width) (height) \\ Area Trapezoid = \frac{1}{2} (base) (h_{1} + h_{2}) \end{aligned}

Insert the appropriate height values $f (x_{i})$ into our chosen method and perform the calculation.

Let’s look at an example.

Example: Approximations using Rectangular and Trapezoidal Areas

Estimate $\int_{1}^{7} x^{2} d x$ using:

Left Endpoint Riemann Sum
Right Endpoint Riemann Sum
Midpoint Rule
Trapezoidal Approximation

with three equal subintervals.

Let’s take a moment to identify the important components of the integral expression.

If $\int_{1}^{7} x^{2} d x$ :

Then the curve is $f (x) = x^{2}$ .
And the closed interval [a,b] is [1,7].

Step 1: Determine the Width

Our width, $Δ x$ , is found by:

$Δ x = \frac{b - a}{n}$

$Δ x = \frac{7 - 1}{3} = \frac{6}{3} = 2 .$

This means that each rectangle or trapezoid will have a width of 2.

Step 2: Applying Approximation Formulas

Next, we’ll use our approximation methods:

For the left endpoint, we calculate are to be: 70

left riemann sum calculation and area under the curve

Notice how the function $f (x)$ is increasing over the interval.

This means that the left Riemann sum is a lower estimation as the rectangles we created are below (under) the curve.

For the right endpoint, it comes out to: 166

right riemann sum calculation and area under the curve

Because $f (x)$ is increasing, the right Riemann sums is an overestimation of the area as the rectangles are above the curve.

Using the midpoint of subintervals we get: 112

midpoint rule calculation and area under the curve

Notice how the rectangles are both under and over the curve $f (x)$ .

This indicates that the midpoint rule is a much better indicator for approximating the true area under the curve.

And with trapezoids, we calculate the area to be: 118

trapezoidal approximation calculation and area under the curve

Since the function $f (x)$ is increasing and concave up, the trapezoidal rule will yield a slightly higher estimation to the true area of the curve.

But it is important to note that the trapezoidal rule, along with the midpoint rule, are the most accurate estimates for approximating the area under the curve.

In general:

For a concave-up function:

The left Riemann sum and midpoint rule underestimate the area.
The right Riemann sum and trapezoidal rule overestimate the area.

For a concave-down function:

The right Riemann sum and trapezoidal rule underestimate the area.
The left Riemann sum and midpoint rule overestimate the area.”

It is always important to check whether a function is increasing or decreasing and its concavity to determine whether an estimation is an over or under approximation.

Isn’t math fascinating? These different approaches shed light on the beauty of approximations!

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