Textbook Question
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.2.58a
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
a. π/2 ≤ x ≤ 3π/2.
Verified step by step guidance1
Identify the function and the interval: The function given is \(y = x \cos x\), and the interval is \(\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}\).
Analyze the graph to determine where the function is above or below the x-axis in the given interval. From the graph, \(y = x \cos x\) is negative between \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) because \(\cos x\) is negative in this interval and \(x\) is positive.
Since the function is below the x-axis in this interval, the area enclosed by the curve and the x-axis is the integral of the absolute value of the function. This means we need to integrate \(-x \cos x\) over \(\left[ \frac{\pi}{2}, \frac{3\pi}{2} \right]\).
Set up the integral for the area: \(\text{Area} = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} -x \cos x \, dx\).
To solve the integral, use integration by parts where you let \(u = x\) and \(dv = \cos x \, dx\). Then compute \(du = dx\) and \(v = \sin x\). Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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. When the function crosses the x-axis, the integral sums positive and negative areas, so the absolute value or splitting the integral is needed to find the total enclosed area.
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Definition of the Definite Integral
Behavior of the Function y = x cos(x)
The function y = x cos(x) oscillates and crosses the x-axis at points where cos(x) = 0, such as x = π/2 and 3π/2. Understanding these zeros helps determine the limits for integration and whether the function is above or below the x-axis in the given interval.
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03:39Integrals of Natural Exponential Functions (e^x)
Splitting the Integral at Zeros of the Function
To find the total area enclosed by the curve and the x-axis when the function changes sign, split the integral at the zeros of the function. Calculate the integral over each sub-interval and take the absolute value before summing to get the total area.
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Related Practice
Textbook Question
Consider the region bounded by the graphs of
y = ln(x), y = 0, and x = e.
a. Find the area of the region.
1
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Textbook Question
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (t³ + 1) dt
Textbook Question
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of (t³ + t) dt
Textbook Question
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 2 of 1/s² ds
Textbook Question
Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.