Riemann sums are a way to estimate the area under a curve. Check out the video for all the deets.
|MPAC 1||Reasoning with definitions and theorems|
|MPAC 2||Connecting concepts|
|MPAC 4||Connecting multiple representations|
|MPAC 5||Building notational fluency|
At the end of the day, Mr. Robin... the Sun Bear and Angora Rabbit's best friend....
...wants to see how popular his friends were by calculating the total number of tickets
sold over 10 hours.
How can we solve for the total number of tickets sold?
Well, let's first take a look at this graph.
It's important to notice that our function is non-negative, which means the function
never outputs a negative y-value.
The y-axis shows the number of tickets sold per hour and the x-axis shows how much time
has passed since opening.
If we multiply y, the tickets sold over time, by x, or time, hours cancel out, so we get
the number of tickets sold at every interval.
This is the same as multiplying the y-value of a point on the curve by the relevant interval
on the x-axis.
If we add up all the intervals, we get the area under the curve.
So, to figure out how many tickets were sold in total, we just need to find the area under
the curve from x = 0 to x = 10.
Unfortunately, the curve is an irregular shape, which means we don't have a formula we can
use to find the exact area. What we can do instead is approximate the
area by drawing a series of rectangles that more or less cover the curve...
...and use the total areas of those rectangles as an estimate of the area under the curve.
There are several different ways we can draw these rectangles, but the most common way
is to put the top left corner of each rectangle directly on the curve.
This will produce a series of rectangles with equal width but varying height based on the
curve, and is called the Left-Hand Sum. For now, we can partition, or slice, the curve
into sub-intervals every two hours, giving us a total of five slices.
As you can see, some of the rectangles drawn underestimate and overestimate the area of
the curve... ...but they basically cancel each other out,
so it gives us a pretty good estimation of the area under the curve.
When finding a left-hand sum, we need to know the value of the function at the left endpoint
of each sub-interval.
Let's look at the first sub-interval between hours 0 and 2 and calculate the area of the
We can see that the left endpoint of the sub-interval is 10.
We know from the good ol' Pre-Algebra days that the area of a rectangle is base times
So we can calculate the area of a rectangle at the first sub-interval by multiplying the
base, or 2, by the height, ten...
...to get 20 as the area of the rectangle. For the second interval, the height is 12,
so the area of the rectangle is two times 12, or 24.
For the third interval, the height is 17, so the area is two times 17, or 34.
The fourth interval has height 23, so the area is two times 23, or 46.
The last interval has height 22, so the area is 2 times 22, or 44.
Adding these up, we find that the total area is approximately 20 plus 24 plus 34 plus 46
plus 44, or 168.
But for people like Mr. Robin, an approximation isn't good enough.
Even if the overestimations and underestimations roughly cancel each other out, the approximation
still isn't exact.
But the more sub-intervals we have, the more accurate our approximation would be.
Suppose we wanted to generalize the width of our subintervals with a formula.
We can label our width with delta x. Since we're dividing the interval... a, b...into...n...equal
sub-intervals, each sub-interval will have length: b minus a over n.
So delta x equals b minus a over n.
The area of a rectangle is length times width.
The length of every rectangle is the height of the curve at each left endpoint...
...so the area can be written as f of x, the length, times delta x, the width.
To find the total area, we can just find the sum of the areas of all the rectangles.
Another way to write this is in sigma notation.
Notice that we are finding the summation of the area of each rectangle, represented by
the formula we calculated earlier.
This form is called a Riemann sum.
Remember how we said that the approximation got more precise with more subintervals?
Let's take this to the extreme and see if we can go from more and more intervals to
an infinite number of intervals. We can take the limit of our Riemann sum as
n approaches infinity, giving us an infinite number of slices.
This will give us an exact approximation of the area under the curve.
Taking this limit as n approaches infinity, gives us a total area of 168.53É
...which is SUPER close if you compare it to the approximation found with rectangles.
Looks like Mr. Robin's friends are pretty popular...with 168 total tickets sold!
Impressive... especially considering that huge Ticketmaster markup.