Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.1.45

Capitolo 8, Problema 8.1.45

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

Guida verificata passo dopo passo

First, identify the curves that bound the region: the upper curve is given by \(y = 2 \cos x\) and the lower curve by \(y = \sec x\) over the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\).

Next, set up the integral for the area between the two curves. The area \(A\) is given by the integral of the difference between the upper and lower functions over the given interval: \(A = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( 2 \cos x - \sec x \right) \, dx\).

Before integrating, verify that \(2 \cos x\) is indeed greater than \(\sec x\) on the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\) to ensure the correct order of subtraction for the area calculation.

Proceed to integrate each term separately: \(\int 2 \cos x \, dx\) and \(\int \sec x \, dx\). Recall that the integral of \(\cos x\) is \(\sin x\), and the integral of \(\sec x\) is \(\ln |\sec x + \tan x|\).

Finally, evaluate the definite integral by substituting the limits \(x = \frac{\pi}{4}\) and \(x = -\frac{\pi}{4}\) into the antiderivative expression, and subtract the results to find the area.

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:

Concetti chiave

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

Definite Integrals for Area Calculation

Definite integrals compute the net area between a curve and the x-axis over a specific interval. When finding the area between two curves, the integral of the difference of the functions over the interval gives the bounded area.

Video consigliato:

Percorso guidato

05:43

Definition of the Definite Integral

Understanding and Comparing Trigonometric Functions

The problem involves trigonometric functions y = 2 cos x and y = sec x. Knowing their behavior, values, and relative positions on the interval helps determine which function is above or below, essential for setting up the integral correctly.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Interval and Domain Considerations

The interval −π/4 ≤ x ≤ π/4 restricts the domain for integration. It's important to verify that both functions are defined and continuous on this interval to ensure the integral is valid and the area calculation is accurate.

Video consigliato:

Percorso guidato

5:10

Finding the Domain and Range of a Graph

Pratica correlata

Domanda del libro di testo

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

views

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.

∫ dx / (x - √x)

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 0 to π/2 of (cot θ dθ)

views

Domanda del libro di testo

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = c * x * √(25 - x²) over [0, 5]

views

Domanda del libro di testo

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

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 2 to ∞ of (dx / √(x² - 1))

views