Table of ValuesThis is one of the most annoying problems to do with integration. The nonuniform change in value makes it difficult to easily get a slope to find the next values in the table. This beautiful video made by a man with a beautiful voice will show you how to integrate when you are thrown in a dark room with a table of values that you need to solve in order to live.
But a table of values isn't a function Mr. MooreWith no function given along with a table of values and their corresponding X's, we are unable to find the exact area underneath the curves. However, we are able to come pretty close using a fancy concept called Riemann Sums. Usually, the method that will bring you closest to the correct value will be the Trapezoid Method. The Trapezoid Method uses areas of trapezoids that are small enough that their slope almost matches the slope of the line exactly. The areas of the trapezoids are then added together and then the total area is evaluated.
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For further information on methods of approximation, follow this link for a detailed explaination of Simpson's Rule. http://mathworld.wolfram.com/SimpsonsRule.html
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