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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.3.63a
8장, 문제 8.3.63a

63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.

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Step 1: Begin by analyzing the integral ∫[0 to π] cos^(2m+1)(x) dx. The integrand is cos^(2m+1)(x), which is an odd power of the cosine function. Recall that cosine is an even function, meaning cos(-x) = cos(x). However, raising cosine to an odd power makes the overall function odd, as odd powers of cosine reverse the sign when x is replaced with -x.
Step 2: Recall the property of definite integrals for odd functions. If f(x) is an odd function and the limits of integration are symmetric about zero (e.g., from -a to a), then ∫[-a to a] f(x) dx = 0. However, in this case, the limits of integration are from 0 to π, which are not symmetric about zero. This means the odd function property does not directly apply here.
Step 3: To determine whether the integral evaluates to zero, consider the behavior of cos^(2m+1)(x) over the interval [0, π]. The cosine function is positive on [0, π/2] and negative on [π/2, π]. Raising cosine to an odd power preserves the sign of the function, meaning cos^(2m+1)(x) is positive on [0, π/2] and negative on [π/2, π].
Step 4: Split the integral into two parts: ∫[0 to π/2] cos^(2m+1)(x) dx and ∫[π/2 to π] cos^(2m+1)(x) dx. The first integral represents the positive contribution, while the second integral represents the negative contribution. However, these contributions do not necessarily cancel out because the intervals [0, π/2] and [π/2, π] are not symmetric, and the magnitude of the function may differ across these intervals.
Step 5: Conclude that the statement 'If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0' is false. Provide a counterexample by evaluating the integral for a specific value of m (e.g., m = 1) to show that the integral does not equal zero. This demonstrates that the odd power of cosine does not lead to cancellation over the interval [0, π].

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. In this case, the integral of cos^(2m+1)(x) from 0 to π is evaluated to determine if it equals zero. Understanding how to compute definite integrals and the properties of the integrand is crucial for solving the problem.
추천 영상:
05:43
Definition of the Definite Integral

Properties of the Cosine Function

The cosine function is periodic and symmetric, specifically even, meaning cos(-x) = cos(x). For odd powers of cosine, such as cos^(2m+1)(x), the function exhibits symmetry about the y-axis, which can lead to cancellation of areas under the curve over symmetric intervals. This property is essential for determining the value of the integral.
추천 영상:
06:21
Properties of Functions

Odd and Even Functions

An odd function satisfies the condition f(-x) = -f(x), while an even function satisfies f(-x) = f(x). The function cos^(2m+1)(x) is odd because it is raised to an odd power. When integrating an odd function over a symmetric interval like [0, π], the result is zero, which is key to answering the question.
추천 영상:
06:21
Properties of Functions
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교과서 질문

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

a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.

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교과서 질문

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].

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교과서 질문

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

a. Graph the position function. At what times does the oscillator pass through the position s = 0?

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교과서 질문

Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.

b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.

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교과서 질문

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.

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교과서 질문

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

67. Let f(x) = √(x³ + 1).

a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.

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