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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.4.45a
Capitolo 5, Problema 5.4.45a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If ƒ is symmetric about the line 𝓍 = 2 , then ∫₀⁴ ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.

Guida verificata passo dopo passo
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Step 1: Begin by understanding the symmetry of the function ƒ about the line 𝓍 = 2. A function is symmetric about a vertical line if, for every point (𝓍, ƒ(𝓍)) on the graph, there exists a corresponding point (4 - 𝓍, ƒ(𝓍)) that mirrors it across the line 𝓍 = 2.
Step 2: Use the property of symmetry to analyze the integral. If ƒ is symmetric about 𝓍 = 2, then the area under the curve from 𝓍 = 0 to 𝓍 = 2 is equal to the area under the curve from 𝓍 = 2 to 𝓍 = 4. This implies that the total integral from 𝓍 = 0 to 𝓍 = 4 can be expressed as twice the integral from 𝓍 = 0 to 𝓍 = 2.
Step 3: Write the integral expression mathematically. Using symmetry, we can state: ∫₀⁴ ƒ(𝓍) d𝓍 = ∫₀² ƒ(𝓍) d𝓍 + ∫₂⁴ ƒ(𝓍) d𝓍. Since the function is symmetric about 𝓍 = 2, ∫₂⁴ ƒ(𝓍) d𝓍 = ∫₀² ƒ(𝓍) d𝓍.
Step 4: Substitute the equality derived from symmetry into the original integral expression. This gives: ∫₀⁴ ƒ(𝓍) d𝓍 = ∫₀² ƒ(𝓍) d𝓍 + ∫₀² ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.
Step 5: Conclude that the statement is true based on the symmetry of the function about the line 𝓍 = 2. The integral from 𝓍 = 0 to 𝓍 = 4 is indeed twice the integral from 𝓍 = 0 to 𝓍 = 2.

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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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Properties of Functions

Definite Integrals

A definite integral, represented as ∫ₐᵇ 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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Definition of the Definite Integral

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 ∫₀⁴ f(x) dx = 2 ∫₀² f(x) dx.
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Properties of Functions
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