Textbook Question
Evaluating integrals Evaluate the following integrals.
β«ββΒ² (3πβ΄β2π + 1) dπ
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.R.35
Find the intervals on which Ζ(π) = β«βΒΉ (tβ3) (tβ6)ΒΉΒΉ dt is increasing and the intervals on which it is decreasing.
Verified step by step guidance1
Step 1: Recognize that the function Ζ(π) is defined as a definite integral with a variable upper limit. To determine where Ζ(π) is increasing or decreasing, we need to compute its derivative using the Fundamental Theorem of Calculus.
Step 2: Apply the Fundamental Theorem of Calculus, which states that if Ζ(π) = β«βΒΉ g(t) dt, then Ζ'(π) = -g(π). Here, g(t) = (t - 3)(t - 6)ΒΉΒΉ, so Ζ'(π) = -(π - 3)(π - 6)ΒΉΒΉ.
Step 3: Analyze the sign of Ζ'(π) to determine where Ζ(π) is increasing or decreasing. Ζ(π) is increasing when Ζ'(π) > 0 and decreasing when Ζ'(π) < 0. This requires solving the inequality -(π - 3)(π - 6)ΒΉΒΉ > 0 and -(π - 3)(π - 6)ΒΉΒΉ < 0.
Step 4: Examine the critical points of Ζ'(π). The factors (π - 3) and (π - 6)ΒΉΒΉ determine the behavior of Ζ'(π). Note that (π - 6)ΒΉΒΉ is always non-negative because it is raised to an odd power. The sign of Ζ'(π) depends on the factor -(π - 3).
Step 5: Determine the intervals of increase and decrease by testing the sign of Ζ'(π) in the intervals divided by the critical points π = 3 and π = 6. Summarize the intervals where Ζ'(π) > 0 (increasing) and Ζ'(π) < 0 (decreasing).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval gives the net change of the function. This theorem allows us to evaluate the integral and find the function's behavior based on its derivative.
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Fundamental Theorem of Calculus Part 1
Derivative and Increasing/Decreasing Functions
A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if its derivative is negative. By analyzing the sign of the derivative, we can determine where the function is rising or falling, which is essential for solving the given problem regarding the intervals of increase and decrease.
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Critical Points
Critical points occur where the derivative of a function is zero or undefined. These points are crucial for determining the intervals of increase and decrease, as they can indicate potential local maxima or minima. By evaluating the derivative at these points, we can ascertain the behavior of the function around them.
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Related Practice
Textbook Question
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(b) β«ββ΄ Ζ(π) dπ
1
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Textbook Question
Evaluate the following derivatives.
d/dπ β«βα΅Λ£ cos tΒ² dt
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Textbook Question
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(a) β«ββ΄ Ζ(π) dπ
1
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Textbook Question
Area functions and the Fundamental Theorem Consider the function
Ζ(t) = { t if β2 β€ t < 0
tΒ²/2 if 0 β€ t β€ 2
and its graph shown below. Let F(π) = β«ββΛ£ Ζ(t) dt and G(π) = β«ββΛ£ Ζ(t) dt.
(e) Evaluate F ''(β1) and F ''(1). Interpret these values.
Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(c) β«βββ΄ (4Ζ(π) β 3g(π))dπ