Skip to main content
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 5.3.89a
Capitolo 5, Problema 5.3.89a

Working with area functions Consider the function ƒ and its graph.
(a) Estimate the zeros of the area function A(𝓍) = ∫₀ˣ ƒ(t) dt , for 0 ≤ 𝓍 ≤ 10 .


<IMAGE>

Guida verificata passo dopo passo
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 \(0 \leq \mathcal{x} \leq 10\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Area Function and Definite Integral

The area function A(x) = ∫₀ˣ 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.
Video consigliato:
Percorso guidato
05:43
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.
Video consigliato:
05:06
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.
Video consigliato:
03:37
Finding Area Between Curves that Cross on the Interval Example 3
Pratica correlata
Domanda del libro di testo

Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).

(a) Find and graph the area function A (𝓍) = ∫ₐˣ ƒ(t) dt .

ƒ(t) = 2t + 5 , a = 0

105
views
Domanda del libro di testo

Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).                                                                                           

                                                                                                                                                                                     

 (a) Find and graph the area function A (𝓍) = ∫ₐˣ ƒ(t) dt .                                                                                                                               

                                                                                                                                                                               

 <IMAGE>                                                                                                                                                                                                           

                                                                                                                                                                                     

 ƒ(t) = 4t + 2 , a = 0

88
views
Domanda del libro di testo

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

(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.

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

51
views
Domanda del libro di testo

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

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

110
views
Domanda del libro di testo

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

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(a) ∫₀^π/2 (2 sin θ ― cos θ) dθ

82
views
Domanda del libro di testo

The velocity in ft/s of an object moving along a line is given by v = ƒ(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)                                                                                                             

                                                                                                                                                                                                

67
views