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
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Odd Functions Odd functions are negatives of each other, on opposite sides of the
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M408C - Notes - Integrating Even and Odd Functions - Even...

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