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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 5.3.87b
Chapter 5, Problem 5.3.87b

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 relationship between the function Ζ’(t) and the area function A(x). The area function A(x) = βˆ«β‚€Λ£ Ζ’(t) dt represents the accumulated area under the curve of Ζ’(t) from t = 0 to t = x. The shape of A(x) depends on the behavior of Ζ’(t).
Step 2: Analyze the graph of Ζ’(t) in image (b). The function Ζ’(t) is a linear function that starts positive at t = 0, decreases linearly, crosses the t-axis, and becomes negative. This indicates that the area accumulated initially increases, then decreases as the negative contributions dominate.
Step 3: Match the behavior of Ζ’(t) to the corresponding area function A(x). Since Ζ’(t) starts positive, A(x) initially increases. As Ζ’(t) crosses the t-axis and becomes negative, A(x) starts decreasing. This matches the graph labeled (A), which shows an initial increase followed by a decrease.
Step 4: Verify the match by considering the slope of A(x). The slope of A(x) is given by Ζ’(t). In graph (A), the slope of A(x) is positive initially (matching the positive Ζ’(t)), becomes zero at the peak (where Ζ’(t) = 0), and then becomes negative (matching the negative Ζ’(t)).
Step 5: Conclude that the graph of Ζ’(t) in image (b) corresponds to the area function A(x) in graph (A). This is based on the consistent relationship between the function and its integral.

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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 the curve of a function f(t) from a specified lower limit to an upper limit. It is denoted as A(x) = βˆ«β‚€Λ£ f(t) dt, where A(x) gives the accumulated area from 0 to x. Understanding this concept is crucial for relating the function f(t) to its area function A(x).
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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 lower limit to x. It provides a way to visualize how the area under the curve changes as x varies. The shape and behavior of the area function graph can reveal important characteristics about the original function f(t), such as its increasing or decreasing nature.
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Finding Area When Bounds Are Not Given

Graphical Interpretation

Graphical interpretation involves analyzing the visual representation of functions and their integrals. For the functions f(t) and their corresponding area functions A(x), understanding how the graph of f(t) influences the graph of A(x) is essential. For instance, if f(t) is positive, A(x) will increase, while if f(t) is negative, A(x) will decrease, illustrating the relationship between the two.
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Related Practice
Textbook Question

Area functions for the same linear function Let Ζ’(t) = t and consider the two area functions A(𝓍) = βˆ«β‚€Λ£ Ζ’(t) dt and F(𝓍) = βˆ«β‚‚Λ£ Ζ’(t) dt .

(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 β‰₯ 2. 

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Textbook Question

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


b) Verify that A'(𝓍) = Ζ’(𝓍).



Ζ’(t) = 4t + 2 , a = 0

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Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (b) ∫ sec 5𝓍 tan 5𝓍 d𝓍

Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(b) Use geometry to find the displacement of the object between t = 0 and t = 2.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          

                                                                                                                                                                                     (b) Suppose Ζ’ is a negative increasing function, for 𝓍 > 0 . Then the area function A(𝓍) = βˆ«β‚€Λ£ Ζ’(t) dt is a decreasing function of 𝓍 .

Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

f(𝓍) = xΒ³ on [-1,2]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

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