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**Starbucks Coffee Company Total Product Sales**

An exploration of the nation’s coffee obsession through Integrals and Riemann Sums

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**Frappuccino Base (Thousands of Pounds)**

Basic Info Starbucks has done some of the legwork for you! Lucky you! They know the total thousands of pounds of base consumed nationally for each month of the past year and have used a graphing calculator to find the function that expresses this information. The function is: f(x)=-.568x2+7.6x-5.068 Month 1 2 3 4 5 6 7 8 9 10 11 12 Frappuccino Base (Thousands of Pounds) 13 16 19 20 21 17 14

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**Starbucks Needs Your Help!**

It’s the end of the year, and Starbucks is getting ready to order Frappuccino Base from it’s suppliers for the upcoming year. Therefore they need to know the average pounds of Frappuccino Base they process and sell over the course of the year. They can always order more or less base as the year progresses, but they need an approximation. As the Calculus Guru you are, Starbucks Coffee Company has turned to you and asked you to help them order more Frappuccino Base.

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Graph The data is presented in a graph like the one on the right. Knowing the graphical representation of this information will help your understanding of the Integral and Riemann Sum process As you can see, the consumption of Frappuccino Base reaches it’s peak in the summer months, maybe because it’s hot??

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What is an Integral? An integral takes a function, such as f(x) and looks at it as if it were the function f’(x). Then by taking the integral we find the original function that f(x) is the derivative of. For example, we have the function f(x)=x2 Say we take the integral of f(x) from 1 to 12, so we write it as 1∫12 (x2)dx Then we undo this to find the function that f(x) is the derivative of, which we write as (1/3)x3]112 Then we plug in f(12) and subtract f(1) from it, like so: (1/3)(12)3 – (1/3)(1)3 This gives us , so 1∫12 (x2)dx=

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**So How Does That Relate to This Problem?**

Integrals are a way to find the total area under a curve or the total change over time. In this situation, an integral would express the total thousands of pounds of Frappuccino Base consumed nationally. Now, there are a few ways to take an integral – you can find the integral, or take Left-hand, Right-hand, Midpoint, or Trapezoid Riemann sums

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**Method 1: Taking an Integral**

f(x)= -.568x2+7.6x-5.068 1∫12 (-.568x2+7.6x-5.068)dx -.189x3+3.8x x]112 [-.189(12)3+3.8(12) (12)] – [-.189(1)3+3.8(1) (1)] thousands of pounds of Frappuccino Base consumed Original function The integral of f(x) from 1 to 12 Take the integral of f(x) Plug in x=12 and x=1 Subtract f(12)-f(1) Find the answer! (this is only an approximation because rounding may influence the actual answer)

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**Method 2: Left-Hand Riemann Sums**

Now, for this we are going to take the Left-Hand Riemann Sum from 1 to 12, splitting the graph into 6 segments Notice that the red lines create 6 rectangles whose left-hand corner intersects with points on the graph. You will be taking the area of each of these rectangles and adding them together.

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**Frappuccino Base (Thousands of Pounds)**

The first rectangle’s width (along the x-axis) is 2, and it’s height or length is 2 (refer to the chart for exact heights) Each rectangle has a width of 2 and a length that is specified by the chart, except for the last rectangle, who’s height is 1 Month 1 2 3 4 5 6 7 8 9 10 11 12 Frappuccino Base (Thousands of Pounds) 13 16 19 20 21 17 14

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**Frappuccino Base (Thousands of Pounds)**

Now, using the area formula (A=W * L), we find the area of each rectangle. Remember, the width is 2 for each rectangle except the last one, and the heights are highlighted in red in the chart. 2( )+10=154 Month 1 2 3 4 5 6 7 8 9 10 11 12 Frappuccino Base (Thousands of Pounds) 13 16 19 20 21 17 14

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**Important Riemann Sum Secrets**

The area under the curve using the Left-Hand Riemann Sum is 154 This means Starbucks consumed 154 millions of pounds of Frappuccino base in this year. Now, because the rectangles do not exactly fill the graph completely, the Left-Hand Riemann Sum is an underestimate of the actual area. The Left-Hand, Right-Hand, and Midpoint Riemann sums follow the same method. All of these are either over or underestimates.

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**Guess what?! Calculators can do all of this for you!**

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Integrals Now that you know how to do this the hard way… calculators can help you do this much easier! First, put the function f(x)= -.568x2+7.6x into Y1 To find the integral of f(x) from 1 to 12, return to the home screen, press math, 9, vars, go to y-vars, then press 1 twice, then include “x,1,12)” and press enter. The answer, , is the integral of f(x) from 1 to 12.

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Riemann Sums To find a Riemann Sum with a calculator, you’ll have to find and enter a program called “ALLSUMS,” a Google search should find it for you. Once the program is loaded, press PRGM, go to ALLSUMS, press enter twice, enter 1 for A,12 for B, then for N put in the number of segments you need to find The answers on the right of the screen correspond with the letters on the top of the screen L (Left-Hand) goes with the first number, R (Right-Hand) goes with the second number, and so on. T is for Trapezoids and M is for Midpoint

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You did it!! You now know how to find the integral of a function using the integral method and Riemann Sums, and you know the shortcuts too! Oh and you found that Starbucks consumes thousands of pounds of Frappuccino base. They’d probably like to know that so they can order more.* *This was a fictional situation. Please don’t call Starbucks.

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