Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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2:08 minutes

Problem 8.3.63b

Textbook Question

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

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

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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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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

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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.

Key Concepts

Definite Integrals

Properties of the Sine Function

Even and Odd Functions

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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.