Textbook Question
{Use of Tech} Areas of regions Find the area of the region π bounded by the graph of Ζ and the π-axis on the given interval. Graph Ζ and show the region π .
Ζ(π) = 2 β |π| on [ β 2 , 4]
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 5.3.5
The linear function Ζ(π) = 3 β π is decreasing on the interval [0, 3]. Is its area function for Ζ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Verified step by step guidance1
Step 1: Understand the problem. The linear function Ζ(π) = 3 - π is given, and we are tasked to determine whether its area function (integral) is increasing or decreasing on the interval [0, 3]. The area function represents the accumulated area under the curve of Ζ(π) starting from the left endpoint 0.
Step 2: Recall the relationship between a function and its area function. The area function is the integral of Ζ(π) with respect to π. Mathematically, the area function A(π) is defined as: . The derivative of the area function, A'(π), is equal to Ζ(π).
Step 3: Analyze the behavior of Ζ(π) on the interval [0, 3]. The function Ζ(π) = 3 - π is linear with a negative slope (-1), meaning it decreases as π increases. Specifically, Ζ(π) starts at 3 when π = 0 and decreases to 0 when π = 3.
Step 4: Determine the behavior of the area function A(π). Since A'(π) = Ζ(π), the area function A(π) increases wherever Ζ(π) is positive. On the interval [0, 3], Ζ(π) is positive (above the x-axis), so the area function A(π) is increasing throughout this interval.
Step 5: Visualize the problem. Draw the graph of Ζ(π) = 3 - π, which is a straight line decreasing from (0, 3) to (3, 0). Shade the area under the curve from π = 0 to a variable endpoint π. As π moves from 0 to 3, the shaded area grows, confirming that the area function A(π) is increasing on [0, 3].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Decreasing Functions
A function is considered decreasing on an interval if, for any two points x1 and x2 within that interval, where x1 < x2, the function value at x1 is greater than the function value at x2 (Ζ(x1) > Ζ(x2)). In this case, the linear function Ζ(π) = 3 - π decreases as x increases, indicating that as we move from 0 to 3, the output values of the function get smaller.
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Area Function
The area function A(x) associated with a function Ζ(π) represents the accumulated area under the curve of Ζ from a starting point (in this case, 0) to a variable endpoint x. Mathematically, it is defined as A(x) = β«[0,x] Ζ(t) dt. The behavior of the area function depends on the values of the original function; if Ζ is decreasing, the area function will reflect this change.
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Relationship Between Function and Area Function
The relationship between a function and its area function is governed by the Fundamental Theorem of Calculus. If the original function is decreasing, the area function will increase at a decreasing rate. This means that while the area function A(x) is increasing as x moves from 0 to 3, the rate of increase diminishes because the heights of the rectangles (representing area) are getting smaller as the function value decreases.
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Related Practice
Textbook Question
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Textbook Question
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« π cosΒ²πΒ² dπ
Textbook Question
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Textbook Question
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
β«α΅ββ Ζ(g(π)) dπ
Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βαΆ |Ζ(π)| dπ
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