M408C - Notes - Integrating Even and Odd Functions

# M408C - Notes - Integrating Even and Odd Functions - Even...

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Even Functions Even and Odd Even functions f ( x ) are the same on the left and right side of the axes; they can’t tell the diﬀerence between + x and - x . In symbols, f is even if f (+ x )= f ( - x ). One example is f ( x x 2 , because f ( - x )=( - x ) 2 =( - 1) 2 ( x ) 2 = x 2 = f ( x ). Below, the graphs of some even functions: Graphs of y = x 2 cos x, y = x 2 1+ x 4 ,y = x 2 - 1 x 2 +1

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Integrating Even Functions One of the things that you see from the graphs is that the sum, diﬀerence, product and quotient of even functions is even. Since even functions are the same on either side of the axis, area is the same on either side. In symbols, Z 0 - a f ( x ) dx = Z a 0 f ( x ) dx This gives the formula, if f is an even function, Z a - a f ( x ) dx =2 Z a 0 f ( x ) dx For example, Z 1 - 1 x 2 - x 2 3 +1 dx Z 1 0 x 2 - x 2 3 dx x 3 3 - 3 5 x 5 3 + x 1 0 1 3 3 - 3 5 1 5 3 - 2[0]=2 4 3 - 3 5 This computation was a lot easer to do because the contribution from the ”0” was simply 2[0] = 0. 0.2 0.4 0.6 0.8 y -0.5 0 0.5 x
Odd Functions Odd functions are negatives of each other, on opposite sides of the
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## This note was uploaded on 03/22/2008 for the course M 408c taught by Professor Mcadam during the Fall '06 term at University of Texas.

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M408C - Notes - Integrating Even and Odd Functions - Even...

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