Change of Variable

Change of Variable - October 3, 2010 1 1. Integration by...

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Unformatted text preview: October 3, 2010 1 1. Integration by substitution: An integral often simlifies when we can denote a part of the function as a new variable (y ). Suppose x = g (y ) where g is a strictly monotone function, then subsitute x ´b using g (y ), a f (x)dx can be calculated by ˆ ˆ b g −1 (b) f (x)dx = f (g (y ))d(g (y )) g −1 (a) a (a) g must be a strictly monotone function here; (b) The limits of integration change from “a to b” to “g −1 (a) to g −1 (b)”. Note that, we don’t need b ≥ a or g −1 (b) ≥ g −1 (a) by using the convention that ˆ ˆ b f (x)dx = − a a f (x)dx b (c) Consider the following integral ˆ 2 xe−2x dx. 0 Let y = 2x, then x = y/2. By substitution (substitute x by y/2), and adjusting the integration limites, the integral becomes ˆ2 ˆ4 ˆ 1 4 −y 1 1 − 5e−4 −2x −y xe dx = (y/2)e d(y/2) = ye dy = (−ye−y −e−y )|4 = . 0 40 4 4 0 0 2. Exercises: √ ´∞ 2 2 (a) We know that 0 e−x /2 dx = 2 π , use the integration by substitution technique, show that, for any σ > 0 and µ, ˆ∞ (x−µ)2 1 √ e− 2σ2 dx = 1/2. 2πσ µ (b) Evaluate ´4 0 2 /4 xex dx. ...
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This note was uploaded on 02/01/2012 for the course STAT 330B taught by Professor Zhou during the Spring '11 term at Iowa State.

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