Textbook Question
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.
(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)
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8. Definite Integrals
Introduction to Definite Integrals
2:20 minutesProblem 5.4.12
Textbook Question
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫²⁰⁰₋₂₀₀ 2x⁵ dx
Verified step by step guidance1
Recognize that the integral ∫²⁰⁰₋₂₀₀ 2x⁵ dx involves a symmetric interval [-200, 200] and an odd function. A function f(x) is odd if f(-x) = -f(x).
Verify that 2x⁵ is an odd function. Substitute -x into the function: f(-x) = 2(-x)⁵ = -2x⁵, which confirms that f(x) = 2x⁵ is odd.
Recall the property of definite integrals: If f(x) is odd and the interval of integration is symmetric about the origin, i.e., [-a, a], then ∫₋ₐₐ f(x) dx = 0.
Apply this property to the given integral ∫²⁰⁰₋₂₀₀ 2x⁵ dx. Since the function is odd and the interval is symmetric, the integral evaluates to 0.
Conclude that symmetry simplifies the computation, and no further calculation is needed for this integral.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Symmetry in Functions
Symmetry in functions refers to the property where a function exhibits even or odd symmetry. An even function, such as f(x) = x², satisfies f(x) = f(-x), while an odd function, like f(x) = x³, satisfies f(x) = -f(-x). Recognizing these properties can simplify the evaluation of integrals, especially over symmetric intervals.
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Definite Integrals
A definite integral calculates the area under a curve between two specified limits. It is represented as ∫[a,b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. Understanding how to evaluate definite integrals is crucial for applying symmetry, as it allows for the simplification of calculations when the function's behavior is known over the interval.
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Properties of Integrals
The properties of integrals include linearity, additivity, and the ability to change limits. For instance, the integral of an even function over a symmetric interval can be simplified to twice the integral from 0 to the upper limit. These properties are essential for efficiently evaluating integrals, particularly when leveraging symmetry to reduce computation.
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