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Published byLorraine Martin Modified over 2 years ago

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A Riemann sum is a method for approximating the total area underneath a curve on a graph. This method is also known as taking an integral. There are 3 forms of Riemann Sums: Left, Right, and Middle.

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B A To find the intervals needed, use the formula: Where B = the upper limit, A = the lower limit, and N = the number of rectangles used. N = 4

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Then incorporate the previous intervals into the formula:

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For a Left Riemann, use all of the functions except for the last one. The Left Riemann under approximates the area under the curve.

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For a Right Riemann, use all of the functions except for the last one. The Right Riemann over approximates the area under the curve.

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For a Middle Riemann, average all the intervals found and plug the averages into the functions. The Middle Riemann is the closest approximation.

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The Middle Riemann is the closest approximation

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N = 4

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1.Click the “PRGM” button. 2.Select the RIEMANN program. 3.Enter your f(x). 4.Enter Lower & Upper bounds. 5.Enter Partitions 6.Select Left, Right, or Midpoint Sum

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Like Riemann Sums, Trapezoidal Rule approximates the are under the curve using trapezoids instead of rectangles to better approximate.

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Use the same formula to find your intervals. Then plug your intervals into the equation:

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N = 4

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Remember to multiply all intervals by 2, excluding the first and last interval.

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N = 4

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1.Click the “PRGM” button. 2.Select the RIEMANN program. 3.Enter your f(x). 4.Enter Lower & Upper bounds. 5.Enter Partitions 6.Select Trapezoid Sum

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Simpson’s rule, created by Thomas Simpson, is the most accurate approximation of the area under a curve as it uses quadratic polynomials instead of rectangles or trapezoids.

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Simpson’s Rule can ONLY be used when there are an even number of partitions. Still use the formula: to find your intervals to plug into the equation.

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N = 4

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When using Simpson’s Rule, multiply all intervals excluding the first and the last alternately between 4 & 2, always starting with 4

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1.Click the “PRGM” button. 2.Select the SIMPSON program. 3.Enter Lower & Upper bounds. 4.Enter your N/2 Partitions. 5.Enter your f(x)

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Section 3.2 – Calculating Areas; Riemann Sums

Section 3.2 – Calculating Areas; Riemann Sums

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