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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.90c
Capitolo 5, Problema 5.3.90c

Working with area functions Consider the function ƒ and its graph.
(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.


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Understand that the function A(x) represents the area function defined as the integral of f(t) from 0 to x, i.e., \(A(x) = \int_0^x f(t) \, dt\). This means A(x) accumulates the net area under the curve of f(t) from 0 to x.
Identify the intervals where f(t) is positive and where it is negative by looking at the graph. When f(t) is above the t-axis, the area contributes positively to A(x), and when f(t) is below the t-axis, the area contributes negatively.
Start sketching A(x) at x=0 with A(0) = 0, since the integral from 0 to 0 is zero. As x increases, the slope of A(x) at any point x is given by f(x), because \(A'(x) = f(x)\).
Use the shape of f(t) to determine the slope of A(x): where f(t) is positive, A(x) is increasing; where f(t) is negative, A(x) is decreasing. Also, where f(t) has local maxima or minima, A(x) will have points where the slope changes accordingly.
Sketch A(x) by accumulating the net area: when f(t) is above the axis, A(x) rises; when f(t) is below, A(x) falls. The graph of A(x) will be smooth and continuous, reflecting the integral of the oscillating function f(t).

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Area Function and Definite Integral

An area function A(x) represents the accumulated area under the curve of a function f(t) from a fixed point (usually 0) to x. It is defined as A(x) = ∫₀ˣ f(t) dt, capturing the net area, which can be positive or negative depending on whether f(t) is above or below the t-axis.
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Percorso guidato
05:43
Definition of the Definite Integral

Relationship Between a Function and Its Area Function

The derivative of the area function A(x) is the original function f(x), i.e., A'(x) = f(x). This means the slope of the graph of A at any point x equals the value of f at x, guiding how the area function increases or decreases based on f's sign and magnitude.
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05:23
Finding Area Between Curves on a Given Interval

Sketching the Area Function Without a Y-Scale

When sketching A(x) without a y-scale, focus on qualitative features: where A(x) increases or decreases (based on f's sign), where it has local maxima or minima (where f crosses zero), and the concavity (related to f's slope). The graph of A(x) is smooth and accumulates area, reflecting the integral of f.
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13:20
Summary of Curve Sketching Example 1
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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.


(c) ∫₄⁰ 6𝓍(4 ― 𝓍) d(𝓍)

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

(c) 1² + 2² + 3² + 4²

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

                                                                                                                                                                    

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

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Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       

                                                                                                                                                                                     c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ? 

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Domanda del libro di testo

Working with area functions Consider the function ƒ and the points a, b, and c.

(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .

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

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Domanda del libro di testo

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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