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.

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.