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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 5.3.94a
Chapter 5, Problem 5.3.94a

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€

Verified step by step guidance
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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 using the Fundamental Theorem of Calculus. The given function is Ζ’(𝓍) = sin(𝓍), and the lower limit of integration is a = 0.
Step 2: Recall the Fundamental Theorem of Calculus. It states that if Ζ’(t) is continuous on [a, x], then the derivative of the area function A(𝓍) with respect to 𝓍 is equal to Ζ’(𝓍). In other words, A'(𝓍) = Ζ’(𝓍).
Step 3: To find A(𝓍), integrate Ζ’(t) = sin(t) with respect to t from the lower limit a = 0 to the upper limit x. The integral of sin(t) is -cos(t). Therefore, A(𝓍) = -cos(t) evaluated from t = 0 to t = x.
Step 4: Apply the limits of integration. Substitute the upper limit x and the lower limit 0 into the expression for A(𝓍). This gives A(𝓍) = [-cos(x)] - [-cos(0)]. Simplify the expression using the fact that cos(0) = 1.
Step 5: The area function A(𝓍) is now expressed in terms of x. You can use this function to evaluate the area for specific values of x, such as b = Ο€/2 and c = Ο€, by substituting these values into A(𝓍).

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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 with integration. It states that if a function is continuous on an interval, then the integral of its derivative over that interval can be computed using the values of the function at the endpoints. This theorem allows us to evaluate definite integrals and find area functions effectively.
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Fundamental Theorem of Calculus Part 1

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated as the limit of Riemann sums and provides a numerical value that corresponds to the total accumulation of the function's values between the two bounds. In this context, it is used to find the area function A(x) from the function f(t).
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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 constant lower limit a to a variable upper limit x. It represents the accumulated area under the curve of f(t) from a to x. In this problem, A(x) = βˆ«β‚Λ£ f(t) dt will help us determine the area under the curve of sin(x) from 0 to x, which is essential for understanding the behavior of the function over the specified interval.
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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 .                                                                                                                               

                                                                                                                                                                               

 <IMAGE>                                                                                                                                                                                                           

                                                                                                                                                                                     

 Ζ’(t) = 4t + 2 , a = 0

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

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

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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