2.02.4.Literacy - Accumulation Authors: Bill Davis, Horacio...

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Accumulation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 2.02 Breaking the Code of the Integral: The Fundamental Formula LITERACY What you should be able to handle away from the machine. · L.1) When you calculate Ÿ 0 t 4 Sin A x 3 E x numerically for various values of t, you get different numbers depending on what t you go with: Clear @ x D ; t = 0.5; NIntegrate A 4 Sin A x 3 E , 8 x, 0, t < , AccuracyGoal Ø 2 E 0.0624349 t = 1.0; NIntegrate A 4 Sin A x 3 E , 8 x, 0, t < , AccuracyGoal Ø 2 E 0.935381 t = 1.5; NIntegrate A 4 Sin A x 3 E , 8 x, 0, t < , AccuracyGoal Ø 2 E 2.34753 The upshot: When you put f @ t D = Ÿ 0 t 4 Sin A x 3 E x, you can plot f @ t D as a function of t: 0.5 1.0 1.5 2.0 t 0.5 1.0 1.5 2.0 f @ t D Even though you don't have a clean formula for f @ t D , you can give a clean formula for f £ @ t D . Do it. · L.2) When you start with f @ x D = Sin @ x D and remember that Sin @ 0 D = 0 and Sin A p 2 E = 1, you can easily write down the value of Ÿ 0 p 2 f £ @ x D x= Ÿ 0 p 2 Cos @ x D x. Do it. · L.3) When you feed a formula for a function g @ x D into Mathematica and ask Mathematica to calculate Ÿ a t g @ x D x, Mathematica looks for a function f @ t D with f £ @ t D = g @ t D and then Mathematica reports Ÿ a t g @ x D x = f @ t D - f @ a D . Once f @ t D with f £ @ t D = g @ t D is found, then the reported answer is guaranteed to be correct because the fundamental formula says that Ÿ a t g @ x D x = Ÿ a t f £ @ x D x = f @ t D - f @ a D . When you calculate integrals, you do the same thing.
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This note was uploaded on 10/11/2011 for the course MATH 241 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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2.02.4.Literacy - Accumulation Authors: Bill Davis, Horacio...

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