Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(a) ∫₋₄⁴ ƒ(𝓍) d𝓍

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Step 1: Understand the symmetry properties of even and odd functions. An even function satisfies ƒ(𝓍) = ƒ(-𝓍), meaning it is symmetric about the y-axis. An odd function satisfies g(𝓍) = -g(-𝓍), meaning it is symmetric about the origin.

Step 2: Recall the property of definite integrals for even functions. If ƒ(𝓍) is even, then ∫₋ₐₐ ƒ(𝓍) d𝓍 = 2∫₀ₐ ƒ(𝓍) d𝓍. This property will be used to evaluate ∫₋₄⁴ ƒ(𝓍) d𝓍.

Step 3: Substitute the given value of ∫₀⁴ ƒ(𝓍) d𝓍 = 10 into the formula for even functions. Using the property, ∫₋₄⁴ ƒ(𝓍) d𝓍 = 2∫₀⁴ ƒ(𝓍) d𝓍.

Step 4: Simplify the expression by multiplying the given value of ∫₀⁴ ƒ(𝓍) d𝓍 by 2. This will give the result for ∫₋₄⁴ ƒ(𝓍) d𝓍.

Step 5: Conclude that the integral ∫₋₄⁴ ƒ(𝓍) d𝓍 depends entirely on the symmetry property of the even function and the given value of ∫₀⁴ ƒ(𝓍) d𝓍.

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주요 개념

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Even Functions

An even function is defined by the property that ƒ(−x) = ƒ(x) for all x in its domain. This symmetry about the y-axis implies that the area under the curve from -a to 0 is equal to the area from 0 to a. Therefore, when integrating an even function over a symmetric interval, the integral can be simplified to twice the integral from 0 to a.

Odd Functions

An odd function satisfies the condition g(−x) = −g(x) for all x in its domain. This property indicates that the function is symmetric about the origin, leading to the conclusion that the integral of an odd function over a symmetric interval around zero is zero. Thus, when evaluating the integral of an odd function from -a to a, the contributions from the negative and positive sides cancel each other out.

Definite Integrals and Symmetry

Definite integrals represent the net area under a curve between two points. When evaluating integrals of even and odd functions over symmetric intervals, the properties of these functions allow for simplifications. For even functions, the integral from -a to a can be expressed as twice the integral from 0 to a, while for odd functions, the integral from -a to a equals zero, highlighting the importance of symmetry in calculus.

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.(a) ∫₋₄⁴ ƒ(𝓍) d𝓍