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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 After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3 ft/sec. 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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Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve:

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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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