Textbook Question
Sigma notation Evaluate the following expressions.
(c) 4
β ΞΊΒ²
ΞΊ=1
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.3.51c
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(c) β«ββ° 6π(4 β π) d(π)
Verified step by step guidance1
Step 1: Recognize that the integral given in the problem, β«ββ° 6π(4 β π) dπ, is related to the integral β«ββ΄ 3π(4 β π) dπ = 32. Notice the limits of integration are reversed, and the integrand has been scaled by a factor of 2.
Step 2: Use the property of integrals that states reversing the limits of integration changes the sign of the integral. Specifically, β«βα΅ f(π) dπ = -β«α΅β f(π) dπ. Apply this property to rewrite β«ββ° 6π(4 β π) dπ as -β«ββ΄ 6π(4 β π) dπ.
Step 3: Factor out the constant 6 from the integral using the property of integrals that allows constants to be factored out. This gives -6 β«ββ΄ π(4 β π) dπ.
Step 4: Recognize that β«ββ΄ π(4 β π) dπ is equivalent to the given integral β«ββ΄ 3π(4 β π) dπ divided by 3, since the integrand in the given integral is scaled by a factor of 3. Therefore, β«ββ΄ π(4 β π) dπ = 32 / 3.
Step 5: Substitute β«ββ΄ π(4 β π) dπ = 32 / 3 into the expression -6 β«ββ΄ π(4 β π) dπ to find the value of the integral. Simplify the expression to complete the solution.
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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. It is denoted as β«βα΅ f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval [a, b]. Understanding this concept is crucial for evaluating integrals and applying properties related to limits.
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Properties of Integrals
The properties of integrals, such as linearity, additivity, and the reversal of limits, are essential for simplifying and evaluating integrals. For instance, the linearity property states that β«(c * f(x)) dx = c * β«f(x) dx for a constant 'c'. Additionally, the property of reversing limits states that β«βα΅ f(x) dx = -β«α΅β f(x) dx. These properties allow for manipulation of integrals to facilitate easier computation.
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Substitution in Integrals
Substitution is a technique used to simplify the evaluation of integrals by changing the variable of integration. This method involves selecting a new variable 'u' that simplifies the integrand, allowing for easier integration. For example, if u = g(x), then dx can be expressed in terms of du, transforming the integral into a more manageable form. Mastery of substitution is vital for solving complex integrals effectively.
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Related Practice
Textbook Question
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(c) β«ββΆ (3Ζ(π) β g(π)) dπ
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.
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Textbook Question
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(c) 1Β² + 2Β² + 3Β² + 4Β²
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Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .
Ζ(π) = β 12π (πβ1) (πβ 2) ; a = 0 , b = 1 , c = 2
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Textbook Question
Approximating areas Estimate the area of the region bounded by the graph of Ζ(π) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.
(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.
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