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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.2.62a
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 guidance
1
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:
07:33
Euler'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:
05:06
Finding 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?

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 3 of (2x - 1) dx

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
views
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

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.