Textbook Question
7β64. Integration review Evaluate the following integrals.
12. β« from -5 to 0 of dx / β(4 - x)
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8. Definite Integrals
Introduction to Definite Integrals
6:48 minutesProblem 5.4.45a
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If Ζ is symmetric about the line π = 2 , then β«ββ΄ Ζ(π) dπ = 2 β«βΒ² Ζ(π) dπ.
Verified step by step guidance1
Step 1: Begin by understanding the symmetry of the function Ζ about the line π = 2. A function is symmetric about a vertical line if, for every point (π, Ζ(π)) on the graph, there exists a corresponding point (4 - π, Ζ(π)) that mirrors it across the line π = 2.
Step 2: Use the property of symmetry to analyze the integral. If Ζ is symmetric about π = 2, then the area under the curve from π = 0 to π = 2 is equal to the area under the curve from π = 2 to π = 4. This implies that the total integral from π = 0 to π = 4 can be expressed as twice the integral from π = 0 to π = 2.
Step 3: Write the integral expression mathematically. Using symmetry, we can state: β«ββ΄ Ζ(π) dπ = β«βΒ² Ζ(π) dπ + β«ββ΄ Ζ(π) dπ. Since the function is symmetric about π = 2, β«ββ΄ Ζ(π) dπ = β«βΒ² Ζ(π) dπ.
Step 4: Substitute the equality derived from symmetry into the original integral expression. This gives: β«ββ΄ Ζ(π) dπ = β«βΒ² Ζ(π) dπ + β«βΒ² Ζ(π) dπ = 2 β«βΒ² Ζ(π) dπ.
Step 5: Conclude that the statement is true based on the symmetry of the function about the line π = 2. The integral from π = 0 to π = 4 is indeed twice the integral from π = 0 to π = 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Symmetry in Functions
A function is symmetric about a vertical line, such as x = 2, if for every point (a, f(a)) on the graph, there is a corresponding point (4-a, f(a)). This means that the function's values are mirrored across the line x = 2, which can affect the evaluation of integrals over symmetric intervals.
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Definite Integrals
A definite integral, represented as β«βα΅ f(x) dx, calculates the area under the curve of the function f(x) from x = a to x = b. The properties of definite integrals, including linearity and the ability to split intervals, are crucial for evaluating integrals over symmetric intervals and understanding how symmetry impacts the integral's value.
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Properties of Integrals and Symmetry
When a function is symmetric about a vertical line, the area under the curve from one side of the line can be related to the area from the other side. Specifically, if f is symmetric about x = 2, then the integral from 0 to 4 can be expressed as the sum of two equal integrals from 0 to 2, leading to the relationship β«ββ΄ f(x) dx = 2 β«βΒ² f(x) dx.
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