Multiple Choice
Let . What is the average value of on the interval ?
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8. Definite Integrals
Average Value of a Function
1:30 minutesProblem 5.RE.15a
Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(a) β«βββ΄ Ζ(π) dπ
Verified step by step guidance1
Step 1: Understand the symmetry properties of even and odd functions. An even function satisfies Ζ(π) = Ζ(-π), meaning it is symmetric about the y-axis. An odd function satisfies g(π) = -g(-π), meaning it is symmetric about the origin.
Step 2: Recall the property of definite integrals for even functions. If Ζ(π) is even, then β«βββ Ζ(π) dπ = 2β«ββ Ζ(π) dπ. This property will be used to evaluate β«βββ΄ Ζ(π) dπ.
Step 3: Substitute the given value of β«ββ΄ Ζ(π) dπ = 10 into the formula for even functions. Using the property, β«βββ΄ Ζ(π) dπ = 2β«ββ΄ Ζ(π) dπ.
Step 4: Simplify the expression by multiplying the given value of β«ββ΄ Ζ(π) dπ by 2. This will give the result for β«βββ΄ Ζ(π) dπ.
Step 5: Conclude that the integral β«βββ΄ Ζ(π) dπ depends entirely on the symmetry property of the even function and the given value of β«ββ΄ Ζ(π) dπ.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even Functions
An even function is defined by the property that Ζ(βx) = Ζ(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.
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Odd Functions
An odd function satisfies the condition g(βx) = βg(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.
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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.
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