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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 8.2.57c
Capitolo 8, Problema 8.2.57c

Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
c. 2π ≤ x ≤ 3π.

Guida verificata passo dopo passo
1
Identify the function and the interval: The function given is \(y = x \sin x\), and we need to find the area enclosed by this curve and the x-axis for \(2\pi \leq x \leq 3\pi\).
Analyze the behavior of the function on the interval: From the graph, observe that \(y = x \sin x\) is positive on the interval \(2\pi \leq x \leq 3\pi\) because \(\sin x\) is positive in this interval and \(x\) is positive as well.
Set up the integral for the area: Since the function is above the x-axis in this interval, the area is given by the definite integral of the function from \(2\pi\) to \(3\pi\):
\[\text{Area} = \int_{2\pi}^{3\pi} x \sin x \, dx\]
To solve this integral, use integration by parts where you let \(u = x\) and \(dv = \sin x \, dx\). Then compute $du = dx$ and \(v = -\cos x\). Apply the integration by parts formula and evaluate the resulting expression at the bounds \(2\pi\) and \(3\pi\) to find the area.

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Definite Integral for Area Calculation

The definite integral of a function over an interval gives the net area between the curve and the x-axis. To find the total area enclosed, you integrate the absolute value of the function or split the integral at points where the function crosses the x-axis.
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Percorso guidato
05:43
Definition of the Definite Integral

Behavior of the Function y = x sin(x)

The function y = x sin(x) oscillates due to the sine component, with amplitude increasing linearly because of the x multiplier. Understanding where the function crosses the x-axis (roots) helps determine intervals where the function is positive or negative, which is crucial for area calculation.
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Integrals of Natural Exponential Functions (e^x)

Handling Areas Below the x-axis

When the curve lies below the x-axis, the definite integral yields a negative value. To find the actual area, you take the absolute value of the integral over those intervals or split the integral at zeros and sum the absolute values of each part.
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05:23
Finding Area Between Curves on a Given Interval
Pratica correlata
Domanda del libro di testo

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 2 of x dx

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Domanda del libro di testo

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

c. u = arctan x

What is the value of the integral?

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Domanda del libro di testo

89. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / x², y = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.

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Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 0 to π of sin(t) dth

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Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 0 to 2 of (t³ + t) dt

27
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Domanda del libro di testo

90. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / √x, y = 0, x = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.

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