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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 5.3.22b
Chapter 5, Problem 5.3.22b

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

Verified step by step guidance
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Step 1: Understand the problem. The goal is to verify that the derivative of the area function A(x), which represents the area under the curve of f(t) = 4t + 2 from a = 0 to x, is equal to the function f(x). This involves using the Fundamental Theorem of Calculus.
Step 2: Define the area function A(x). The area function A(x) is given by the definite integral of f(t) from a = 0 to x. Mathematically, this is expressed as: Ax=∫0xf(t)dt
Step 3: Apply the Fundamental Theorem of Calculus. According to the theorem, if A(x) is defined as the integral of f(t) from a constant to x, then the derivative of A(x) with respect to x is equal to f(x). Mathematically, this is expressed as: A'(x)=f(x)
Step 4: Substitute the given function f(t) = 4t + 2 into the Fundamental Theorem of Calculus. This means that the derivative of A(x) should equal f(x), which is 4x + 2.
Step 5: Conclude the verification. By the Fundamental Theorem of Calculus, the derivative of the area function A(x) is indeed equal to f(x). Therefore, A'(x) = f(x) = 4x + 2, verifying the statement.

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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 starting point 'a' to a variable endpoint 'x'. In this context, it quantifies the area between the x-axis and the function f(t) = 4t + 2 over the interval [a, x]. Understanding this concept is crucial for analyzing how the area changes as 'x' varies.
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Finding Area When Bounds Are Not Given

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes a connection between differentiation and integration. It states that if A(x) is the area function defined as the integral of f(t) from a to x, then the derivative A'(x) equals f(x). This theorem is essential for verifying the relationship A'(x) = f(x) in the given problem.
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Fundamental Theorem of Calculus Part 1

Derivative

A derivative measures how a function changes as its input changes, representing the slope of the tangent line to the function at a given point. In this case, A'(x) indicates the rate of change of the area function A(x) with respect to x. Understanding derivatives is key to verifying that A'(x) equals the function f(x) in the context of the problem.
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Derivatives
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

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (b) ∫ sec 5𝓍 tan 5𝓍 d𝓍

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.



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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

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

(b) Find the mass of the right half of the rod (5 ≀ x ≀ 10) .

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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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