Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.RE.15c

Chapter 5, Problem 5.RE.15c

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.

(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍

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Step 1: Recall the symmetry properties of even and odd functions. An even function satisfies ƒ(𝓍) = ƒ(−𝓍), and its integral over a symmetric interval [−a, a] is twice the integral over [0, a]. An odd function satisfies g(𝓍) = −g(−𝓍), and its integral over a symmetric interval [−a, a] is 0.

Step 2: Break the given integral ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍)) d𝓍 into two separate integrals: ∫₋₄⁴ 4ƒ(𝓍) d𝓍 and ∫₋₄⁴ −3g(𝓍) d𝓍. This uses the linearity property of integrals.

Step 3: For the first term, ∫₋₄⁴ 4ƒ(𝓍) d𝓍, note that ƒ(𝓍) is an even function. Therefore, ∫₋₄⁴ ƒ(𝓍) d𝓍 = 2∫₀⁴ ƒ(𝓍) d𝓍. Multiply this result by 4 to account for the coefficient.

Step 4: For the second term, ∫₋₄⁴ −3g(𝓍) d𝓍, note that g(𝓍) is an odd function. The integral of an odd function over a symmetric interval [−a, a] is 0. Therefore, this term evaluates to 0.

Step 5: Combine the results from Step 3 and Step 4. The final integral ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍)) d𝓍 simplifies to the result obtained from the first term, as the second term is 0.

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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

An even function satisfies the property f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, an odd function satisfies g(-x) = -g(x), indicating symmetry about the origin. These properties are crucial for evaluating integrals over symmetric intervals, as they allow simplifications based on the behavior of the functions.

Properties of Definite Integrals

Definite integrals have specific properties that can simplify calculations. For instance, the integral of an even function over a symmetric interval [-a, a] is twice the integral from 0 to a, while the integral of an odd function over the same interval is zero. These properties help in evaluating integrals without direct computation.

Linear Combination of Integrals

The linearity of integrals allows us to combine integrals of functions through addition and scalar multiplication. Specifically, ∫(af(x) + bg(x))dx = a∫f(x)dx + b∫g(x)dx, where a and b are constants. This property is essential for evaluating integrals involving multiple functions, as it enables the separation of terms for easier computation.

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Related Practice

Textbook Question

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.

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Textbook Question

Function defined by an integral Let ƒ(𝓍) = ∫₀ˣ (t ― 1)¹⁵ (t―2)⁹ dt .

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Textbook Question

Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n:

∫₀¹ 𝓍ⁿd𝓍 + ∫₀¹ ⁿ√(𝓍d𝓍) = 1

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Textbook Question

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.

(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.

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Textbook Question

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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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.(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍

Verified video answer for a similar problem:

Key Concepts

Even and Odd Functions

Properties of Definite Integrals

Linear Combination of Integrals

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.

(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍