Textbook Question
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
f(x) = x + 1 on [0,4]; n = 4
(d) Calculate the left and right Riemann sums.
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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 5.3.51d
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.
(d) β«ββΈ 3π(4 β π) d(π)
Verified step by step guidance1
Step 1: Recognize that the integral β«ββΈ 3π(4 β π) dπ can be split into two parts: β«ββ΄ 3π(4 β π) dπ and β«ββΈ 3π(4 β π) dπ, based on the interval of integration.
Step 2: Use the given information that β«ββ΄ 3π(4 β π) dπ = 32 to evaluate the first part of the integral.
Step 3: For the second part, β«ββΈ 3π(4 β π) dπ, consider the symmetry of the function 3π(4 β π). Analyze whether the function changes sign or remains symmetric over the interval [4, 8].
Step 4: If the function is symmetric and the integral over [0, 4] is known, use properties of symmetry to determine the integral over [4, 8]. Alternatively, compute β«ββΈ 3π(4 β π) dπ directly by substitution or other methods.
Step 5: Combine the results of the two integrals, β«ββ΄ 3π(4 β π) dπ and β«ββΈ 3π(4 β π) dπ, to find the value of β«ββΈ 3π(4 β π) dπ.
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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].
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Definition 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 a sum is the sum of the integrals, and the integral from a to b can be expressed as the negative of the integral from b to a. These properties allow for simplification and manipulation of integrals to facilitate evaluation.
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Substitution Method
The substitution method is a technique used to simplify the evaluation of integrals by changing the variable of integration. By substituting a new variable, often denoted as u, the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler function.
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Related Practice
Textbook Question
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(d) β«ββΆ (g(π) β f(π) dπ
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Textbook Question
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
Ζ(π) = 2x + 1 on [0,4] ; n = 4
d) Calculate the midpoint Riemann sum.
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Textbook Question
Use Table 5.6 to evaluate the following indefinite integrals.
(d) β« cos π/7 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.
(d) If A(π) = 3πΒ²β πβ 3 is an area function for Ζ, then
B(π) = 3πΒ² β π is also an area function for Ζ.
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Textbook Question
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} Ζ(π) = cos π on [0. Ο/2]; n = 4
(d) Calculate the left and right Riemann sums.
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