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 8.7.35
Capitolo 8, Problema 8.7.35

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π

Guida verificata passo dopo passo
1
Identify the formula for the surface area of a curve revolved around the x-axis. The surface area \( S \) is given by: \[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
For the given curve \( y = \sin x \), determine the interval of integration, which is from \( a = 0 \) to \( b = \pi \).
Compute the derivative of \( y \) with respect to \( x \): \[ \frac{dy}{dx} = \cos x \]
Substitute \( y = \sin x \) and \( \frac{dy}{dx} = \cos x \) into the surface area formula: \[ S = \int_0^{\pi} 2\pi \sin x \sqrt{1 + \cos^2 x} \, dx \]
Set up the integral for evaluation, and then use a calculator or computer to approximate the value of the integral to two decimal places.

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:
6m

Concetti chiave

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

Surface Area of Revolution

The surface area of a solid formed by revolving a curve around an axis is found using an integral formula. For revolution about the x-axis, the formula is S = ∫ 2πy √(1 + (dy/dx)²) dx over the given interval. This calculates the total area of the curved surface generated.
Video consigliato:
09:07
Example 1: Minimizing Surface Area

Derivative of the Function

To apply the surface area formula, you need the derivative dy/dx of the function y = sin x. The derivative, cos x, measures the slope of the curve at each point and is essential for computing the integrand's square root term, which accounts for the curve's steepness.
Video consigliato:
06:30
Derivatives of Other Trig Functions

Definite Integration and Numerical Approximation

Evaluating the surface area requires integrating the function from 0 to π. Since the integral may not have a simple closed form, numerical methods or a calculator are used to approximate the value to two decimal places, ensuring an accurate and practical solution.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ 1/(x(ln(x))²) dx

31
views
Domanda del libro di testo

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)

38
views
Domanda del libro di testo

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

16
views
Domanda del libro di testo

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ √(9 - w²) dw / w²

22
views
Domanda del libro di testo

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))

28
views
Domanda del libro di testo

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x dx) / (25 + 4x²)

22
views