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)
5
views
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas this helpful?
Key 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.
Recommended video:
06:21
Properties of Functions
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.
Recommended video:
05:43Definition of the Definite Integral
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.
Recommended video:
06:21Properties of Functions
5:43mWatch next
Master Definition of the Definite Integral with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice