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 8.3.26
Capitolo 8, Problema 8.3.26

Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) dθ

Guida verificata passo dopo passo
1
Recognize that the integrand is \( \sqrt{1 - \cos^2(\theta)} \). Using the Pythagorean identity, recall that \( \sin^2(\theta) + \cos^2(\theta) = 1 \), so \( 1 - \cos^2(\theta) = \sin^2(\theta) \).
Rewrite the integral using this identity: \( \int_0^{\pi} \sqrt{\sin^2(\theta)} \, d\theta \). Since the square root of a square is the absolute value, this becomes \( \int_0^{\pi} |\sin(\theta)| \, d\theta \).
Analyze the behavior of \( \sin(\theta) \) on the interval \( [0, \pi] \). Note that \( \sin(\theta) \) is non-negative on this interval, so \( |\sin(\theta)| = \sin(\theta) \) for \( \theta \in [0, \pi] \).
Simplify the integral to \( \int_0^{\pi} \sin(\theta) \, d\theta \). Now, find the antiderivative of \( \sin(\theta) \), which is \( -\cos(\theta) \).
Apply the Fundamental Theorem of Calculus by evaluating \( -\cos(\theta) \) from \( 0 \) to \( \pi \), i.e., compute \( [-\cos(\theta)]_0^{\pi} = -\cos(\pi) + \cos(0) \).

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.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that 1 - cos²(θ) equals sin²(θ) simplifies the integral significantly, allowing the square root to be expressed as |sin(θ)|.
Video consigliato:
7:17
Verifying Trig Equations as Identities

Absolute Value in Integrals

When integrating expressions involving square roots of squared functions, the result is the absolute value of the original function. Since √(sin²(θ)) = |sin(θ)|, understanding how to handle absolute values over the interval [0, π] is essential, as sin(θ) changes sign within this range.
Video consigliato:
Percorso guidato
06:37
Average Value of a Function

Definite Integration over Piecewise Functions

Definite integrals involving absolute values often require splitting the integral at points where the function inside the absolute value changes sign. For sin(θ) on [0, π], it is positive on [0, π/2] and positive on [π/2, π], so the integral can be evaluated by considering these intervals separately or by using symmetry.
Video consigliato:
Percorso guidato
05:36
Piecewise Functions
Pratica correlata
Domanda del libro di testo

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₋₁³ (4x² - 7) / (2x + 3) dx

27
views
Domanda del libro di testo

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀⁴ dx / √(4 − x)

29
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–22.

∫₀^π 8cos⁴(2πx) dx

26
views
Domanda del libro di testo

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4

33
views
Domanda del libro di testo

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x arctan(x) dx

43
views
Domanda del libro di testo

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).

84
views