# Riemann & Trapezoidal Sums

## Approximating Riemann Integrals

Introduction to approximation techniques other than vertical strips (Trapezoid Rule and Simpson's Rule).

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## The Trapezoid Rule

Derivation of the Trapezoid Rule for approximating Riemann integrals.

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Definition of this rule for approximating the area under a curve, including a measure of the error for this method compared to the actual value of the area.

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Estimating a definite integral of the sine-squared function using three intervals of the Trapezoidal Rule.

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## Simpson's Rule

Derivation of Simpson's Rule for approximating Riemann integrals. Worked example using Trapezoid Rule and Simpson's Rule.

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Definition and formula for this rule which uses quadratics to approximate the area under a curve, including a comparison of this and the Trapezoid Rule. Also includes an applet for finding the area under a curve using the rectangular left, rectangular right, trapezoid, and Simpson's Rule.

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## Special Riemann Sums

Finding Riemann sums with fixed widths using the leftmost, rightmost, maximum, and minimum argument in each strip, including comparisons of each and an applet for finding the leftmost and rightmost Riemann sums for a function.

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## Extrapolations and Better Approximations

Using extrapolation to greatly improve the accuracy of approximations using the Trapezoid Rule or Simpson's Rule.

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## Riemann Sum

Computing the right endpoint Riemann sum of an integral and then using that answer to evaluate a limit.

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## Riemann Sums and Integrals

Three problems which involve evaluating Riemann sums and integrals.

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## Trapezoid Rule and Simpsons Rule

Finding the approximate value of an integral using each rule with two subintervals.

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## Trapezoid Rule and Simpson's Rule: Baseball Stats

Estimating the number of hits a player got in a month using the two rules.

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## Numeric Approximations of Integrals

Using Riemann Sums, the Trapezoid Rule, and Simpson's Rule to approximate a definite integral.

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## Definite Integrals

Seven questions which involve using sigma notation for sums, computing Riemann sums for definite integrals, and evaluating limits by relating them to Riemann sums.

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## Numerical Integration

Five questions which involve approximating integrals using Riemann sums, the Trapezoidal Rule, and Simpson's Rule.

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Applet which uses the Left Hand Rule, Right Hand Rule, Trapezoid Rule, and Simpson's Rule to approximate the area under a specified curve.

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