Textbook Question
Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.
(b) ∫₃⁶ (―3g(𝓍)) d𝓍
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8. Definite Integrals
Introduction to Definite Integrals
1:50 minutesProblem 5.4.18
Textbook Question
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫₋π/₂^π/² 5 sin θ dθ
Verified step by step guidance1
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.
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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.
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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.
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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.
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