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5.1 Estimating with Finite Sums Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 Greenfield Village, Michigan

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time velocity How far will the object have traveled after 4 seconds? Consider an object moving at a constant rate of 3 ft/sec. What would the graph of this look like? Since rate. time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

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If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. (The units work out.) Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. This is called the Left-hand Rectangular Approximation Method (LRAM). Approximate area:

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We could also use a Right-hand Rectangular Approximation Method (RRAM). Approximate area:

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Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). Approximate area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

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Approximate area: width of subinterval With 8 subintervals: The exact answer for this problem is.

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Inscribed rectangles are all below the curve: Circumscribed rectangles are all above the curve: (Do not have to touch at left corner.) (Do not have to touch at right corner.)

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We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation. The TI-89 calculator can do these rectangular approximation problems. This is of limited usefulness, since we will learn better methods of finding the area under a curve, but you could use the calculator to check your work.

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If you have the calculus tools program installed: Set up the WINDOW screen as follows:

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Select Calculus Tools and press Enter Press APPS Press F3 Press alpha and then enter: Make the Lower bound: 0 Make the Upper bound: 4 Make the Number of intervals: 4 Press Enter and then 1 Note: We press alpha because the screen starts in alpha lock.

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A father who is very much concerned about his son's bad grades in math decides to register him at Concordia. After his first term there, the son brings home his report card: He's getting "A"s in math. The father is, of course, pleased, but wants to know: "Why are your math grades suddenly so good?" "You know", the son explains, "when I walked into the classroom the first day, and I saw that guy on the wall nailed to a plus sign, I knew one thing: This place means business!"

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