riemann sums, trapezoidal rule, & simpson’s rule

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Riemann Sums, Trapezoidal Rule, & Simpsons Rule. By: Carson Smith & Elisha Farley. Riemann Sums. 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. - PowerPoint PPT Presentation

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Riemann Sums, Trapezoidal Rule, & Simpsons Rule

Riemann Sums, Trapezoidal Rule, & Simpsons RuleBy: Carson Smith & Elisha Farley

Riemann SumsA 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.

Left RiemannMiddle RiemannRight RiemannRiemann Sums IllustratedRiemann Sum Formula

BATo 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

Riemann Sum Formula Cont.

Then incorporate the previous intervals into the formula:

Left Riemann Example

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

Right Riemann Example

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

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

Integration Answer

The Middle Riemann is the closest approximation

Try A Left Riemann!N = 4

Left Riemann SolutionN = 4

Riemann Sum Program UsageClick the PRGM button.Select the RIEMANN program.Enter your f(x).Enter Lower & Upper bounds.Enter PartitionsSelect Left, Right, or Midpoint SumTrapezoidal RuleLike Riemann Sums, Trapezoidal Rule approximates the are under the curve using trapezoids instead of rectangles to better approximate.Trapezoidal Rule Illustrated

Trapezoidal Rule FormulaUse the same formula to find your intervals.Then plug your intervals into the equation:

Trapezoidal Rule Example

N = 4

Trapezoidal Rule Example Cont.Remember to multiply all intervals by 2, excluding the first and last interval.

Try This Trapezoidal Rule Problem!

N = 4Trapezoidal Rule Solution

N = 4

Trapezoidal Rule Program UsageClick the PRGM button.Select the RIEMANN program.Enter your f(x).Enter Lower & Upper bounds.Enter PartitionsSelect Trapezoid Sum

Simpsons RuleSimpsons 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.

Simpsons Rule FormulaSimpsons 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.

Simpsons Rule Example

N = 4

Simpsons Rule Example Cont.

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

Try This Simpsons Rule Problem!

Simpsons Rule Solution

Simpsons Rule Program UsageClick the PRGM button.Select the SIMPSON program.Enter Lower & Upper bounds.Enter your N/2 Partitions.Enter your f(x)

1994 AB 6

1994 AB 6 A Solution

1994 AB 6 B Solution

1994 AB 6 C Solution

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