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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 5.3.96a
Chapter 5, Problem 5.3.96a

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) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Verified step by step guidance
1
Step 1: Recall the Fundamental Theorem of Calculus, which states that if A(𝓍) = βˆ«β‚Λ£ Ζ’(t) dt, then A'(𝓍) = Ζ’(𝓍). This means the area function A(𝓍) is the antiderivative of Ζ’(t) evaluated from a to 𝓍.
Step 2: Write down the given function Ζ’(𝓍) = -12𝓍(𝓍 - 1)(𝓍 - 2). To find A(𝓍), we need to integrate Ζ’(t) with respect to t over the interval [a, 𝓍], where a = 0.
Step 3: Substitute Ζ’(t) = -12t(t - 1)(t - 2) into the integral A(𝓍) = βˆ«β‚Λ£ Ζ’(t) dt. This becomes A(𝓍) = βˆ«β‚€Λ£ -12t(t - 1)(t - 2) dt.
Step 4: Expand the polynomial -12t(t - 1)(t - 2) to simplify the integrand. Multiply the terms step by step to get a single polynomial expression in terms of t.
Step 5: Integrate the resulting polynomial term by term with respect to t. Use the power rule for integration, ∫tⁿ dt = (tⁿ⁺¹)/(n+1), and evaluate the definite integral from 0 to 𝓍 to find 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 is essential for finding area functions, as it allows us to express the area under a curve as an integral.
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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 the context of area functions, it is used to compute the area from a starting point 'a' to a variable endpoint '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 constant lower limit 'a' to a variable upper limit 'x'. It quantifies the area under the curve of f(t) from 'a' to 'x'. This function is crucial for understanding how the area changes as 'x' varies, and it is often used in applications involving accumulation and total quantities.
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Related Practice
Textbook Question

Bounds on an integral Suppose Ζ’ is continuous on [a, b] with Ζ’''(𝓍) > 0 on the interval. It can be shown that (b―a) Ζ’ [(a + b) /2] ≀ βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍 ≀ (b―a) [ (Ζ’(a) + Ζ’(b)) /2]                                                         

                                                                                                                                                                               

(a) Assuming Ζ’ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. 

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Textbook Question

Properties of integrals Consider two functions Ζ’ and g on [1,6] such that βˆ«β‚βΆΖ’(𝓍) d𝓍 = 10 and βˆ«β‚βΆg(𝓍) d𝓍 = 5, βˆ«β‚„βΆΖ’(𝓍) d𝓍 = 5 , and βˆ«β‚β΄g(𝓍) d𝓍 = 2. Evaluate the following integrals.


(a) βˆ«β‚β΄ 3f(𝓍) d𝓍

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Textbook Question

The velocity in ft/s of an object moving along a line is given by v = Ζ’(t) on the interval 0 ≀ t ≀ 8 (see figure), where t is measured in seconds.

a) Divide the interval [0,8] into n = 2 subintervals, [0,4] and [4,8]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,8] (see part (a) of the figure)                                                                                                             


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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

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

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

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Textbook Question

The velocity in ft/s of an object moving along a line is given by v = Ζ’(t) on the interval 0 ≀ t ≀ 6 (see figure), where t is measured in seconds.


(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)                                                                                                             

                                                                                                                                                                                                

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