Textbook Question
If ƒ is an even function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 2 ∫₀ᵃ ƒ(𝓍) d𝓍?
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8. Definite Integrals
Average Value of a Function
2:10 minutesProblem 5.4.50
Textbook Question
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍
Verified step by step guidance1
Step 1: Recall the definitions of even and odd functions. An even function satisfies ƒ(𝓍) = ƒ(-𝓍), while an odd function satisfies ƒ(𝓍) = -ƒ(-𝓍). These properties will help us analyze the symmetry of the integrand.
Step 2: Analyze the composition ƒ(p(𝓍)). Since ƒ is an even function, ƒ(p(𝓍)) will depend on the behavior of p(𝓍). Because p is an odd function, p(-𝓍) = -p(𝓍). Substituting this into ƒ, we get ƒ(p(-𝓍)) = ƒ(-p(𝓍)).
Step 3: Use the property of ƒ being even. Since ƒ is even, ƒ(-p(𝓍)) = ƒ(p(𝓍)). Therefore, ƒ(p(𝓍)) satisfies ƒ(p(-𝓍)) = ƒ(p(𝓍)), which means the integrand ƒ(p(𝓍)) is an even function.
Step 4: Apply the symmetry property of definite integrals. For an even function integrated over a symmetric interval [-a, a], the integral simplifies to 2 times the integral from 0 to a. Specifically, ∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍 = 2∫₀ᵃ ƒ(p(𝓍)) d𝓍.
Step 5: Conclude the simplification. Since the integrand is even, the integral can be simplified as described in Step 4. If further evaluation is needed, substitute specific forms of ƒ and p into the integral and proceed with integration techniques.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even and Odd Functions
Even functions satisfy the property f(-x) = f(x) for all x, meaning their graphs are symmetric about the y-axis. Odd functions, on the other hand, satisfy f(-x) = -f(x), indicating symmetry about the origin. Understanding these properties is crucial for analyzing the symmetry of composite functions and their integrals.
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Composite Functions
A composite function is formed when one function is applied to the result of another function, denoted as (f ∘ g)(x) = f(g(x)). When dealing with even and odd functions, the symmetry properties of the outer and inner functions can affect the overall symmetry of the composite function, which is essential for determining the nature of the integrand.
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Integration of Symmetric Functions
When integrating even functions over symmetric intervals, the integral can be simplified to twice the integral from 0 to a, while the integral of odd functions over symmetric intervals equals zero. This property allows for easier computation of integrals involving even and odd functions, particularly in the context of the given problem.
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