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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 5.3.89a
Chapter 5, Problem 5.3.89a

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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1
Recall that the area function \(A(\mathcal{x}) = \int_0^{\mathcal{x}} f(t) \, dt\) represents the net area under the curve \(y = f(t)\) from \(t=0\) to \(t=\mathcal{x}\).
To estimate the zeros of \(A(\mathcal{x})\), we need to find values of \(\mathcal{x}\) where the net signed area from 0 to \(\mathcal{x}\) is zero. This means the positive and negative areas cancel out exactly.
Observe the graph of \(f(t)\): it starts below the \(t\)-axis, crosses the axis near \(t=1\), rises above the axis until about \(t=8\), and then falls below the axis again.
Identify intervals where the area under \(f(t)\) is positive or negative. The area function \(A(\mathcal{x})\) increases when \(f(t) > 0\) and decreases when \(f(t) < 0\). The zeros of \(A(\mathcal{x})\) occur where the accumulated positive and negative areas balance out.
Estimate the points \(\mathcal{x}\) where the total net area is zero by visually comparing the areas above and below the \(t\)-axis on the graph. These points correspond to the zeros of \(A(\mathcal{x})\) within the interval \(0 \leq \mathcal{x} \leq 10\).

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Key Concepts

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

Area Function and Definite Integral

The area function A(x) = βˆ«β‚€Λ£ f(t) dt represents the accumulated area under the curve of f(t) from 0 to x. It measures the net area, counting regions below the t-axis as negative. Understanding this helps estimate where the total accumulated area returns to zero.
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Definition of the Definite Integral

Zeros of the Area Function

Zeros of the area function occur where the accumulated net area equals zero. This means the positive and negative areas under f(t) balance out. Identifying these points requires analyzing where the integral changes sign, often linked to the behavior of f(t) over the interval.
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Finding Area When Bounds Are Not Given

Relationship Between f(t) and A(x)

By the Fundamental Theorem of Calculus, the derivative of the area function A(x) is the original function f(x). This means A(x) increases when f(x) is positive and decreases when f(x) is negative. This relationship helps interpret the graph of f(t) to find where A(x) crosses zero.
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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

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(a) 1 + 2 + 3 + 4 + 5

Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = βˆ«β‚€^Ο€/2 (cos ΞΈ ― 2 sin ΞΈ) dΞΈ = ―1

(a) βˆ«β‚€^Ο€/2 (2 sin ΞΈ ― cos ΞΈ) dΞΈ

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€