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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 5.3.15a
Chapter 5, Problem 5.3.15a

Area functions for constant 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 for Ζ’.
fig
Ζ’(t) = 5 , a = 0

Verified step by step guidance
1
Step 1: Understand the problem. The function Ζ’(t) = 5 is a constant function, and we are tasked with finding the area function A(𝓍) = βˆ«β‚Λ£ Ζ’(t) dt, where a = 0. The graph shows a rectangle with height Ζ’(t) = 5 and width determined by the interval [a, 𝓍].
Step 2: Recall the formula for the definite integral of a constant function. For a constant function Ζ’(t) = c, the integral βˆ«β‚Λ£ Ζ’(t) dt simplifies to c * (𝓍 - a). In this case, c = 5 and a = 0.
Step 3: Substitute the values into the formula. Replace c with 5 and a with 0 in the formula for the definite integral. This gives A(𝓍) = 5 * (𝓍 - 0).
Step 4: Simplify the expression for A(𝓍). The area function becomes A(𝓍) = 5𝓍, which represents the area of the rectangle as a function of 𝓍.
Step 5: Graph the area function A(𝓍). The graph of A(𝓍) = 5𝓍 is a straight line passing through the origin with a slope of 5. This represents how the area under the curve Ζ’(t) = 5 grows linearly as 𝓍 increases.

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Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is denoted as βˆ«β‚Λ£ f(t) dt, where 'a' is the lower limit and 'x' is the upper limit. This concept is fundamental in calculating the total accumulation of quantities, such as area, over an interval.
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Definition of the Definite Integral

Area Function

The area function A(x) is defined as the integral of a function f(t) from a fixed point 'a' to a variable point 'x'. It quantifies the area under the curve of f(t) from 'a' to 'x', providing a way to visualize how the area changes as 'x' varies. In this case, with f(t) = 5, A(x) will yield a linear function.
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Finding Area When Bounds Are Not Given

Constant Function

A constant function is a function that always returns the same value regardless of the input. In this scenario, f(t) = 5 is a constant function, meaning the height of the rectangle representing the area under the curve remains constant. This simplifies the calculation of the area, as it can be computed as the product of the base and height.
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Exponential Functions
Related Practice
Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(a) Describe the motion of the object over the interval [0,6].

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ’, Ζ’', and Ζ’'' are continuous functions for all real numbers.                                                                                                                                                           

                                                                                                                                                                    

(a) βˆ« Ζ’(𝓍) Ζ’'(𝓍) d𝓍 = Β½ (Ζ’(𝓍))Β² + C.

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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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Textbook Question

Sigma notation Evaluate the following expressions.

(a)    10                                                                                                                                                                               

       βˆ‘ ΞΊ                                                                                                                                                                          

       ΞΊ=1                         

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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)

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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€

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