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Riemann Sums A Method For Approximating the Areas of Irregular Regions

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Topics of Discussion The Necessity for Approximation Left Hand Rectangular Approximation Methods Right Hand Rectangular Approximation Methods Midpoint Rectangular Approximation Methods Approximations from Numeric Data Comparing the Methods

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The Necessity for Approximation In previous courses, you’ve learned how to find the areas of regular geometric shapes using various formulas…

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The Necessity for Approximation However, when we have shapes like these, there are no nice, neat formulas with which to calculate their area…

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The Necessity for Approximation To address this issue, we use a large number of small rectangles to approximate the area of one of these irregular regions. We may choose to use rectangles with different widths or rectangles with the same width.

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The Necessity for Approximation Things to Remember When we know a function, it is best to approximate the area using rectangles with the same width. When we only know certain points of the function, we will let those points dictate the widths of the rectangles that we use.

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The BIG Unanswered Question??? We’ve talked about the widths of the rectangles that we will use for approximation, but how will we decide what the heights of these rectangles will be?

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Determining the Heights of our Rectangles Basically, unless specified, we can use any point on the function in the given interval for the height of a rectangle. However, we typically choose to use one of 3 points: The Left Endpoint The Right Endpoint The Midpoint

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Left Hand Rectangular Approximation Method As its name indicates, in the Left Hand Rectangular Approximation Method (LRAM), we will use the value of the function at the left endpoint to determine the heights of the rectangles.

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Left Hand Rectangular Approximation Method

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Right Hand Rectangular Approximation Method As its name indicates, in the Right Hand Rectangular Approximation Method (RRAM), we will use the value of the function at the right endpoint to determine the heights of the rectangles.

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Right Hand Rectangular Approximation Method

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Midpoint Rectangular Approximation Method As its name indicates, in the Midpoint Rectangular Approximation Method (MRAM), we will use the value of the function at the midpoint to determine the heights of the rectangles.

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Midpoint Rectangular Approximation Method

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Approximations from Numeric Data These methods for approximation are most useful when we don’t actually know a function, but where we have several numerical data points that have been collected. In this situation, the widths of our rectangles will be determined for us (and may not all be the same). TimeSpeed 03 38 514 117 1518

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Approximations from Numeric Data TimeSpeed 03 38 514 117 1518 To find the distance traveled in this situation, we can choose to use either LRAM or RRAM (but not MRAM since we don’t know the values at the midpoints. However, unlike our previous examples, the widths of our rectangles will all be different.

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Approximations from Numeric Data TimeSpeed 03 38 514 117 1518 To find LRAM, we will use the function value at the left endpoint and the varying widths of rectangles.

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Approximations from Numeric Data TimeSpeed 03 38 514 117 1518 To find RRAM, we will use the function value at the right endpoint and the varying widths of rectangles.

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Resource Pages http://www.math.ucla.edu/~ronmie ch/Java_Applets/Riemann/ http://www.math.ucla.edu/~ronmie ch/Java_Applets/Riemann/ http://www.synergizedsolutions.co m/simpsons/pictures.shtml http://www.synergizedsolutions.co m/simpsons/pictures.shtml

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Section 7.6 – Numerical Integration. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.

Section 7.6 – Numerical Integration. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.

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