Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.AAE.7
Capitolo 8, Problema 8.AAE.7

Evaluate the limits in Exercise 7 and 8.
lim (x → ∞) ∫₋ˣ^ˣ sin t dt

Guida verificata passo dopo passo
1
Recognize that the problem asks for the limit as \(x\) approaches infinity of the integral \(\int_{-x}^{x} \sin t \, dt\).
Recall the Fundamental Theorem of Calculus, which tells us how to evaluate definite integrals using antiderivatives.
Find the antiderivative of \(\sin t\), which is \(-\cos t\).
Evaluate the definite integral \(\int_{-x}^{x} \sin t \, dt\) by computing \([-\cos t]_{-x}^{x} = -\cos x - (-\cos(-x))\).
Use the fact that cosine is an even function, so \(\cos(-x) = \cos x\), and simplify the expression before taking the limit as \(x \to \infty\).

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:
2m

Concetti chiave

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

Definite Integral and its Geometric Interpretation

A definite integral represents the net area under a curve between two limits. In this problem, the integral of sin(t) from -x to x measures the accumulated area under the sine curve over a symmetric interval, which is essential for understanding how the integral behaves as x approaches infinity.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral

Limit of a Function as x Approaches Infinity

The limit as x approaches infinity examines the behavior of a function or expression when x grows without bound. Here, evaluating lim (x → ∞) ∫₋ˣ^ˣ sin t dt requires understanding how the integral changes as the interval expands indefinitely.
Video consigliato:
06:11
Limits of Rational Functions: Denominator = 0

Properties of the Sine Function and its Integral

The sine function is periodic and oscillates between -1 and 1. Its integral over symmetric intervals often cancels out due to its odd symmetry, which is crucial for evaluating the limit of the integral from -x to x as x tends to infinity.
Video consigliato:
Percorso guidato
06:21
Properties of Functions
Pratica correlata
Domanda del libro di testo

Finding volume

Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

28
views
Domanda del libro di testo

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

42
views
Domanda del libro di testo

Length of a curve

Find the length of the curve

y = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.

38
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–6.

∫ x arcsin x dx

69
views
Domanda del libro di testo

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ cos²(2θ) sin(θ) dθ

29
views
Domanda del libro di testo

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ cos t dt / (1 - cos t)

20
views