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Introduction to Integrals with Riemann Sums, a la Shmoop.

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Tickets to see the Sun Bear and Angora Rabbit are selling out fast.

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They don't do any tricks or anything. They're just hilarious to look at.

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The ticket machine comes with a real-time ticket tracker that graphs the number of tickets

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sold per hour throughout the day.

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

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

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sold over 10 hours.

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

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

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

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never outputs a negative y-value.

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

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has passed since opening.

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

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the number of tickets sold at every interval.

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

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on the x-axis.

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

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

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the curve from x = 0 to x = 10.

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

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

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

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

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

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

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

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

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

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

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

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

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

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of each sub-interval.

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

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rectangle.

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

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

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height.

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

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

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

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

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

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

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

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

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plus 44, or 168.

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

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

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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.

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

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

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So delta x equals b minus a over n.

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The area of a rectangle is length times width.

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

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

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

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Another way to write this is in sigma notation.

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

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the formula we calculated earlier.

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This form is called a Riemann sum.

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

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

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

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

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

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

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

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

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Impressive... especially considering that huge Ticketmaster markup.

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