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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 5.3.22a
Chapter 5, Problem 5.3.22a

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 .                                                                                                                               
                                                                                                                                                                               
 <IMAGE>                                                                                                                                                                                                           
                                                                                                                                                                                     
 Ζ’(t) = 4t + 2 , a = 0

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Step 1: Understand the problem. You are tasked with finding the area function A(𝓍) = βˆ«β‚Λ£ Ζ’(t) dt, where Ζ’(t) = 4t + 2 and a = 0. This involves integrating the given linear function Ζ’(t) with respect to t, starting from the lower limit a = 0 to the upper limit x.
Step 2: Set up the integral. Write the area function as A(𝓍) = βˆ«β‚€Λ£ (4t + 2) dt. This represents the accumulation of the area under the curve Ζ’(t) = 4t + 2 from t = 0 to t = x.
Step 3: Perform the integration. Break the integral into two parts: βˆ«β‚€Λ£ 4t dt and βˆ«β‚€Λ£ 2 dt. Use the power rule for integration on the first term and the constant rule for integration on the second term. For the first term, ∫ 4t dt = 4 * (tΒ² / 2) = 2tΒ². For the second term, ∫ 2 dt = 2t.
Step 4: Apply the limits of integration. Substitute the limits of integration (from t = 0 to t = x) into the result of the integration. For the first term, evaluate 2tΒ² at t = x and t = 0. For the second term, evaluate 2t at t = x and t = 0.
Step 5: Combine the results. After applying the limits, combine the results from both terms to express the area function A(𝓍). The final expression will be a function of x that represents the accumulated area under the curve Ζ’(t) = 4t + 2 from t = 0 to t = x.

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Key Concepts

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

Area Function

An area function, denoted as A(x), represents the accumulated area under a curve from a fixed point 'a' to a variable point 'x'. It is defined mathematically as A(x) = βˆ«β‚Λ£ f(t) dt, where f(t) is the function being integrated. This concept is crucial for understanding how the area changes as 'x' varies, and it provides insights into the behavior of the function over an interval.
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Definite Integral

The definite integral is a fundamental concept in calculus that calculates the net area under a curve between two points on the x-axis. It is represented as βˆ«β‚Λ£ f(t) dt, where 'a' is the lower limit and 'x' is the upper limit. This integral not only quantifies the area but also accounts for the sign of the function, allowing for the determination of positive and negative areas.
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Definition of the Definite Integral

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between variables. For area functions, the graph illustrates how the area accumulates as 'x' increases. Understanding how to graph functions is essential for interpreting the behavior of the area function and analyzing its properties, such as increasing or decreasing trends.
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Related Practice
Textbook Question

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.



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

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

Working with area functions Consider the function Ζ’ and its graph.

(a) Estimate the zeros of the area function A(𝓍) = βˆ«β‚€Λ£ Ζ’(t) dt , for 0 ≀ 𝓍 ≀ 10 .


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Textbook Question

Approximating displacement The velocity in ft/s of an object moving along a line is given by v = 3tΒ² + 1 on the interval 0 ≀ t ≀ 4, where t is measured in seconds.

(a) Divide the interval [0,4] into n = 4 subintervals, [0,1] , [1.2] , [2,3] , and [3,4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 4] (see part (a) of the figure)

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€