Integral Approximations (Trapezium, Mid-Ordinate and Simpson's Rules)

This applet shows three numerical methods for integration - the Trapezium Rule, the Mid-Ordinate Rule and Simpson's Rule. The Trapezium and the Mid-Ordinate Rules are second-order methods (the error decreases in proportion to the square of number of strips used). Simpson's Rule is a weighted average of the other two - (1xTrapezium + 2xMid-Ordinate)/3, and is a fourth-order method (error deceases as the fourth power of the number of strips). It is hard to visualise graphically. For cubic and quadratic graphs it gives an exact value, eqivalent to the value given by algebraic integration. This result can be proved using algebra. A Challenge - prove it!

Author and programmer: Ron Barrow

UK Years 12-13, KS5, AS and A2 Level Mathematics, Numerical Methods

   
           

Instructions below    See also:   Algebraic integration

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How to Use this Applet

This applet is designed to help you understand three very important numerical methods of integration - the Trapezium Rule, the Mid-Ordinate Rule and Simpson's Rule.
Varying A, B, C and D changes the curves. (Note that if you're in quadratic mode, A no longer is relevant, so clicking A+ or A- will have no effect.)
You can move the two integral limits by clicking and dragging the two circles on the x-axis with your mouse.
Checking or unchecking the boxes at the top will display or hide the different rules.

Clicking on "Strips +" or "Strips -" increases or reduces the number of strips taken between the two limits, from 2 to 16 strips.
By playing around you should develop an insight into how the area between the curve and the x-axis is related to the shapes which approximate it. You should also investigate whether and why the approximations are underestimates or overestimates, and how the error can be reduced. The error is given to you by comparing the numerical method with the exact integral (from algebraic calculus), rounded to 2 d.p.