Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.4.18
Chapter 5, Problem 5.4.18

Symmetry in integrals Use symmetry to evaluate the following integrals.
∫₋π/₂^π/² 5 sin θ dθ

Verified step by step guidance
1
Recognize that the integral involves a symmetric interval about zero: [-π/2, π/2]. This symmetry can simplify the evaluation process.
Observe the integrand, 5 sin(θ). The function sin(θ) is odd, meaning sin(-θ) = -sin(θ). This property is crucial for determining the behavior of the integral over symmetric intervals.
Use the property of odd functions: When integrating an odd function over a symmetric interval about zero, the integral evaluates to zero. This is because the positive and negative contributions cancel each other out.
Conclude that the integral ∫₋π/₂^π/₂ 5 sin(θ) dθ equals zero due to the symmetry and the odd nature of sin(θ).
No further computation is needed, as the symmetry property directly provides the result.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Symmetry in Integrals

Symmetry in integrals refers to the property that allows certain integrals to be simplified based on the symmetry of the function being integrated. If a function is even (symmetric about the y-axis), the integral from -a to a can be computed as twice the integral from 0 to a. If the function is odd (symmetric about the origin), the integral over a symmetric interval around zero is zero.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially in integrals involving angles. The sine function, sin(θ), is periodic and has specific properties, such as being odd, which means sin(-θ) = -sin(θ). Understanding these properties is crucial for evaluating integrals that involve trigonometric functions.
Recommended video:
6:04
Introduction to Trigonometric Functions

Definite Integrals

A definite integral calculates the area under a curve defined by a function over a specific interval [a, b]. It is represented as ∫_a^b f(x) dx and provides a numerical value. Evaluating definite integrals often involves techniques such as substitution, integration by parts, or leveraging symmetry, which can simplify the computation significantly.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

44
views
Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₄^π/⁴ sec² x dx

95
views
Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ d𝓍 / (√1 ― 9𝓍²)

92
views
Textbook Question

{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of ƒ and the 𝓍-axis on the given interval. Graph ƒ and show the region 𝑅.                                              

                                                                                                                                                                                    

 ƒ(𝓍) = 𝓍² (𝓍 ― 2) on [ ―1 , 3]

29
views
Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  ∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)  

119
views
Textbook Question

Determine the intervals on which the function g(𝓍) = ∫ₓ⁰ t / (t² + 1) dt  is concave up or concave down.

49
views