# 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 Calculus Integrals Language English Language MPAC 1 Reasoning with definitions and theorems MPAC 2 Connecting concepts MPAC 4 Connecting multiple representations MPAC 5 Building notational fluency MPAC 6 Communicating Math Calculus

### 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.

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

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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.