Skip to main content
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 8.8.89c
Chapter 8, Problem 8.8.89c

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.

Verified step by step guidance
1
Identify the region to be revolved: it is bounded by the curve \(y = \frac{1}{x^{2}}\), the line \(y = 0\), and the vertical line \(x = 1\), all in the first quadrant.
Since the region is revolved about the y-axis, consider using the method of cylindrical shells. The shell radius is the distance from the y-axis, which is \(x\), and the shell height is the vertical distance between the curve and the x-axis, which is \(\frac{1}{x^{2}}\).
Set up the volume integral using the shell method formula: \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\). Here, \(a = 1\) and \(b \to \infty\) because the region extends infinitely to the right.
Write the integral explicitly: \(V = 2\pi \int_{1}^{\infty} x \cdot \frac{1}{x^{2}} \, dx = 2\pi \int_{1}^{\infty} \frac{1}{x} \, dx\).
Evaluate the improper integral by taking the limit as the upper bound approaches infinity, and analyze whether the volume converges or diverges.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Setting up the Region for Integration

Understanding the boundaries of the region is essential. Here, the region is in the first quadrant bounded by y = 1/x², y = 0, and x = 1. Identifying these limits helps determine the interval and shape of the area to be revolved.
Recommended video:
07:45
Area of Polar Regions

Method of Cylindrical Shells for Volume

When revolving a region around the y-axis, the cylindrical shells method is often used. It involves integrating the volume of thin cylindrical shells with radius x, height given by the function, and thickness dx, which simplifies volume calculation.
Recommended video:
04:48
Finding Volume Using Disks

Improper Integrals and Convergence

Since the region extends infinitely in the y-direction, the volume integral is improper. Evaluating whether the integral converges (has a finite value) is crucial to ensure the volume is well-defined and to compute it correctly.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Related Practice
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 1 to 2 of x dx

Textbook Question

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.

1
views
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
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