Textbook Question
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.2.62a
Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about
a. The y-axis.
(See Exercise 57 for a graph.)
Verified step by step guidance1
Identify the region to be revolved: the area bounded by the curve \(y = x \sin(x)\), the x-axis (\(y=0\)), and the vertical lines \(x=0\) and \(x=\pi\).
Since the solid is generated by revolving around the y-axis, use the method of cylindrical shells. The formula for the volume is \(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\).
Determine the radius and height of a typical shell: the radius is the distance from the y-axis, which is \(x\), and the height is the function value \(y = x \sin(x)\).
Set up the integral for the volume: \(V = \int_0^{\pi} 2\pi \cdot x \cdot (x \sin(x)) \, dx = \int_0^{\pi} 2\pi x^2 \sin(x) \, dx\).
Evaluate the integral \(\int_0^{\pi} x^2 \sin(x) \, dx\) using integration by parts twice, then multiply the result by \(2\pi\) to find the volume.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume can be computed using integral methods such as the disk/washer or shell method, depending on the axis of rotation and the shape of the region.
Recommended video:
04:48
Finding Volume Using Disks
Shell Method
The shell method calculates volume by integrating cylindrical shells formed when the region is revolved around an axis. For rotation about the y-axis, vertical slices parallel to the axis create shells with radius and height functions, making it suitable for this problem.
Recommended video:
07:33Euler's Method
Function Behavior and Bounds
Understanding the function y = x sin(x) and its behavior on the interval [0, π] is essential. Knowing the bounds and how the curve interacts with the x-axis helps set up correct integral limits and expressions for radius and height in the volume calculation.
Recommended video:
05:06Finding Area When Bounds Are Not Given
Related Practice
Textbook Question
Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.
a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?
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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 3 of (2x - 1) dx
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
Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.
Textbook Question
Centroid:
Find the centroid of the region cut from the first quadrant by the curve
y = 1/√(x + 1) and the line x = 3.