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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.3.87a
Chapter 5, Problem 5.3.87a

Matching functions with area functions Match the functions Ζ’, whose graphs are given in a― d, with the area functions A (𝓍) = βˆ«β‚€Λ£ Ζ’(t) dt, whose graphs are given in A–D.


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Verified step by step guidance
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Step 1: Understand the problem. We are tasked with matching the graph of the function f(t) (given in the first image) with the graph of its corresponding area function A(x) = βˆ«β‚€Λ£ f(t) dt (given in the second image). The area function A(x) represents the accumulated area under the curve of f(t) from t = 0 to t = x.
Step 2: Analyze the graph of f(t). The graph of f(t) in image (a) is a constant function, where f(t) = c (a positive constant) for all t in the interval [0, b]. This means the value of f(t) does not change with t, and the area under the curve will grow linearly as x increases.
Step 3: Determine the behavior of A(x). Since f(t) is constant, the integral A(x) = βˆ«β‚€Λ£ f(t) dt will result in a linear function. Specifically, A(x) will increase at a constant rate because the area under a constant function is proportional to the width of the interval.
Step 4: Match A(x) with the correct graph. Among the graphs labeled A–D in the second image, the graph labeled (C) shows a straight line, which corresponds to a linear function. This matches the behavior of A(x) derived from the constant function f(t).
Step 5: Conclude the match. The graph of f(t) in image (a) corresponds to the area function A(x) in graph (C). This is because the constant rate of change in f(t) leads to a linear accumulation of area in A(x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral represents the signed area under a curve defined by a function f(t) from a lower limit to an upper limit. It is denoted as βˆ«β‚α΅‡ f(t) dt, where 'a' and 'b' are the bounds of integration. This concept is crucial for understanding how area functions A(x) are derived from the original function f(t) by integrating it over a specified interval.
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Definition of the Definite Integral

Area Function

An area function A(x) is defined as the integral of a function f(t) from a fixed point (usually 0) to a variable upper limit x. Mathematically, it is expressed as A(x) = βˆ«β‚€Λ£ f(t) dt. This function provides a way to visualize how the area under the curve of f(t) accumulates as x changes, which is essential for matching the area functions with their corresponding original functions.
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Finding Area When Bounds Are Not Given

Graphical Interpretation of Integrals

The graphical interpretation of integrals involves visualizing the area under the curve of a function f(t) as the integral is computed. The shape and behavior of the area function A(x) can be analyzed by observing how the area accumulates as x increases. Understanding this relationship helps in matching the graphs of functions with their corresponding area functions, as the characteristics of the area function reflect the properties of the original function.
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Related Practice
Textbook Question

Area functions for linear functions Consider the following functions Ζ’ and real numbers a (see figure).                                                                                           

                                                                                                                                                                                     

 (a) Find and graph the area function A (𝓍) = βˆ«β‚Λ£ Ζ’(t) dt .                                                                                                                               

                                                                                                                                                                               

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 Ζ’(t) = 4t + 2 , a = 0

Textbook Question

Suppose Ζ’ is an odd function, βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 = 3 , and βˆ«β‚€βΈ Ζ’(𝓍) d𝓍 = 9 .


(a) Evaluate βˆ«β‚‹β‚ˆβ΄ Ζ’(𝓍) d𝓍 .

Textbook Question

Properties of integrals Use only the fact that βˆ«β‚€β΄ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(a) βˆ«β‚„β° 3𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Substitutions Suppose Ζ’ is an even function with βˆ«β‚€βΈ Ζ’(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       

(a) βˆ«ΒΉβ‚‹β‚ π“Ζ’(𝓍²) d𝓍

Textbook Question

Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the 𝓍-direction and 2b in the y-direction is (𝓍²/aΒ²) + (yΒ² /bΒ²) = 1.

(a) Let dΒ² denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ ―a, a] to show that the average value of dΒ² is (aΒ² + 2bΒ²) /3 .

Textbook Question

Working with area functions Consider the function Ζ’ and the points a, b, and c.

(a) Find the area function A (𝓍) = βˆ«β‚Λ£ Ζ’(t) dt using the Fundamental Theorem.

Ζ’(𝓍) = sin 𝓍 ; a = 0 , b = Ο€/2 , c = Ο€