Introduction to Integrals with Riemann Sums

Riemann sums are a way to estimate the area under a curve. Check out the video for all the deets.

AP CalculusIntegrals
LanguageEnglish Language
MPAC 1Reasoning with definitions and theorems
MPAC 2Connecting concepts
MPAC 4Connecting multiple representations
MPAC 5Building notational fluency
MPAC 6Communicating
MathCalculus

Transcript

00:25

At the end of the day, Mr. Robin... the Sun Bear and Angora Rabbit's best friend....

00:29

...wants to see how popular his friends were by calculating the total number of tickets

00:34

sold over 10 hours.

00:36

How can we solve for the total number of tickets sold?

00:39

Well, let's first take a look at this graph.

00:41

It's important to notice that our function is non-negative, which means the function

00:46

never outputs a negative y-value.

00:49

The y-axis shows the number of tickets sold per hour and the x-axis shows how much time

00:55

has passed since opening.

00:57

If we multiply y, the tickets sold over time, by x, or time, hours cancel out, so we get

01:03

the number of tickets sold at every interval.

01:05

This is the same as multiplying the y-value of a point on the curve by the relevant interval

01:11

on the x-axis.

01:13

If we add up all the intervals, we get the area under the curve.

01:15

So, to figure out how many tickets were sold in total, we just need to find the area under

01:20

the curve from x = 0 to x = 10.

01:23

Unfortunately, the curve is an irregular shape, which means we don't have a formula we can

01:29

use to find the exact area. What we can do instead is approximate the

01:34

area by drawing a series of rectangles that more or less cover the curve...

01:38

...and use the total areas of those rectangles as an estimate of the area under the curve.

01:43

There are several different ways we can draw these rectangles, but the most common way

01:47

is to put the top left corner of each rectangle directly on the curve.

01:53

This will produce a series of rectangles with equal width but varying height based on the

01:59

curve, and is called the Left-Hand Sum. For now, we can partition, or slice, the curve

02:05

into sub-intervals every two hours, giving us a total of five slices.

02:10

As you can see, some of the rectangles drawn underestimate and overestimate the area of

02:15

the curve... ...but they basically cancel each other out,

02:18

so it gives us a pretty good estimation of the area under the curve.

02:23

When finding a left-hand sum, we need to know the value of the function at the left endpoint

02:28

of each sub-interval.

02:30

Let's look at the first sub-interval between hours 0 and 2 and calculate the area of the

02:36

rectangle.

02:37

We can see that the left endpoint of the sub-interval is 10.

02:41

We know from the good ol' Pre-Algebra days that the area of a rectangle is base times

02:46

height.

02:46

So we can calculate the area of a rectangle at the first sub-interval by multiplying the

02:52

base, or 2, by the height, ten...

02:54

...to get 20 as the area of the rectangle. For the second interval, the height is 12,

03:01

so the area of the rectangle is two times 12, or 24.

03:05

For the third interval, the height is 17, so the area is two times 17, or 34.

03:10

The fourth interval has height 23, so the area is two times 23, or 46.

03:14

The last interval has height 22, so the area is 2 times 22, or 44.

03:21

Adding these up, we find that the total area is approximately 20 plus 24 plus 34 plus 46

03:27

plus 44, or 168.

03:31

But for people like Mr. Robin, an approximation isn't good enough.

03:35

Even if the overestimations and underestimations roughly cancel each other out, the approximation

03:40

still isn't exact.

03:41

But the more sub-intervals we have, the more accurate our approximation would be.

03:48

Suppose we wanted to generalize the width of our subintervals with a formula.

03:56

We can label our width with delta x. Since we're dividing the interval... a, b...into...n...equal

04:04

sub-intervals, each sub-interval will have length: b minus a over n.

04:09

So delta x equals b minus a over n.

04:12

The area of a rectangle is length times width.

04:15

The length of every rectangle is the height of the curve at each left endpoint...

04:19

...so the area can be written as f of x, the length, times delta x, the width.

04:26

To find the total area, we can just find the sum of the areas of all the rectangles.

04:31

Another way to write this is in sigma notation.

04:34

Notice that we are finding the summation of the area of each rectangle, represented by

04:38

the formula we calculated earlier.

04:41

This form is called a Riemann sum.

04:44

Remember how we said that the approximation got more precise with more subintervals?

04:50

Let's take this to the extreme and see if we can go from more and more intervals to

04:55

an infinite number of intervals. We can take the limit of our Riemann sum as

04:59

n approaches infinity, giving us an infinite number of slices.

05:04

This will give us an exact approximation of the area under the curve.

05:14

Taking this limit as n approaches infinity, gives us a total area of 168.53É

05:19

...which is SUPER close if you compare it to the approximation found with rectangles.

05:26

Looks like Mr. Robin's friends are pretty popular...with 168 total tickets sold!

05:30

Impressive... especially considering that huge Ticketmaster markup.