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

검증된 단계별 안내

Step 1: Understand the problem. The statement claims that the integral of sin^m(x) from 0 to π equals 0 for any positive integer m. We need to determine if this is true or false and provide an explanation or counterexample.

Step 2: Recall the properties of the sine function. The sine function, sin(x), is symmetric about π/2 within the interval [0, π]. This symmetry affects the behavior of sin^m(x) depending on whether m is odd or even.

Step 3: Analyze the case when m is odd. For odd values of m, sin^m(x) is an odd function about π/2. This means the integral from 0 to π will not necessarily be zero because the positive and negative contributions do not cancel out.

Step 4: Analyze the case when m is even. For even values of m, sin^m(x) is an even function about π/2. This symmetry ensures that the integral from 0 to π will be positive, as the function remains non-negative throughout the interval.

Step 5: Conclude that the statement is false. The integral ∫[0 to π] sin^m(x) dx is not always zero for positive integer values of m. Provide a counterexample, such as m = 2, where the integral evaluates to a positive value.

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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 sin^m(x) from 0 to π calculates the total area between the curve of sin^m(x) and the x-axis over that interval. Understanding how definite integrals work is crucial for evaluating the truth of the statement.

Properties of the Sine Function

The sine function oscillates between -1 and 1, and its behavior over the interval [0, π] is particularly important. Specifically, sin(x) is non-negative in this interval, meaning that sin^m(x) will also be non-negative for any positive integer m. This property is essential for determining whether the integral can equal zero.

Even and Odd Functions

An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. The function sin^m(x) is even when m is even, and odd when m is odd. This distinction affects the evaluation of the integral, as the integral of an odd function over a symmetric interval around zero is zero, while the integral of an even function is positive, reinforcing the need to analyze the parity of m.

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