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 5.3.102b
Capitolo 5, Problema 5.3.102b

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.


(b) Calculate g'(𝓍)


g(𝓍) = ∫₀ˣ sin (πt² ) dt ( a Fresnel integral) 

Guida verificata passo dopo passo
1
Step 1: Recognize that the function g(𝓍) is defined as a definite integral with a variable upper limit. This is a classic application of the Fundamental Theorem of Calculus, which states that if g(𝓍) = ∫ₐˣ f(t) dt, then g'(𝓍) = f(𝓍), provided f is continuous.
Step 2: Identify the integrand of g(𝓍). In this case, the integrand is sin(πt²). According to the Fundamental Theorem of Calculus, g'(𝓍) will be equal to the integrand evaluated at the upper limit of integration, which is 𝓍.
Step 3: Substitute the upper limit 𝓍 into the integrand. This means g'(𝓍) = sin(π𝓍²).
Step 4: Confirm that the derivative g'(𝓍) does not require further simplification, as the integrand sin(π𝓍²) is already expressed in terms of 𝓍.
Step 5: Note that no additional integration or differentiation is needed, as the problem specifically asks for g'(𝓍), which is directly obtained using the Fundamental Theorem of Calculus.

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:
1m

Concetti chiave

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

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative can be found by evaluating the integrand at that upper limit. Specifically, if g(x) = ∫ₐˣ f(t) dt, then g'(x) = f(x). This theorem is essential for calculating the derivative of functions defined by integrals.
Video consigliato:
Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, between two specified limits. In the context of the given function g(x) = ∫₀ˣ sin(πt²) dt, the integral computes the area under the curve of sin(πt²) from 0 to x. Understanding how to evaluate definite integrals is crucial for applying the Fundamental Theorem of Calculus.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral

Fresnel Integral

The Fresnel integral is a specific type of integral that arises in wave optics and is defined as g(x) = ∫₀ˣ sin(πt²) dt. It is important in various applications, including diffraction and interference patterns. Recognizing the properties and behavior of Fresnel integrals helps in understanding the function g(x) and its derivative.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals
Pratica correlata
Domanda del libro di testo

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.

b) Calculate g'(𝓍)


g(𝓍) = ∫₀ˣ sin² t dt

65
views
Domanda del libro di testo

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.



63
views
Domanda del libro di testo

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.


f(x) = sin 2x on [0,3π/4]


(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

55
views
Domanda del libro di testo

The following functions are positive and negative on the given interval.

ƒ(𝓍) = xe⁻ˣ on [-1,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

74
views
Domanda del libro di testo

{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.


ƒ(𝓍) = x² ― 1 on [2,5] ; n = 75

(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.

88
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].

81
views