Textbook Question
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?
1
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 8.2.57c
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.
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
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. 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.
Recommended video:
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.
Recommended video:
03:39Integrals 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.
Recommended video:
05:23Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question
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.
1
views
Textbook Question
Consider the region bounded by the graphs of y = sin⁻¹(x), y = 0, and x = 1/2.
b. Find the centroid of the region.
Textbook Question
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
Textbook Question
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 -2 to 0 of (x² - 1) dx
Textbook Question
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.
1
views