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 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 8.2.61b
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about
b. The line x = π/2.
Verified step by step guidance1
Identify the region to be revolved: it is bounded by the x-axis (y=0), the y-axis (x=0), and the curve \(y = \cos(x)\) for \(0 \leq x \leq \frac{\pi}{2}\), all in the first quadrant.
Since the solid is generated by revolving the region about the vertical line \(x = \frac{\pi}{2}\), use the method of cylindrical shells to find the volume.
Set up the volume integral using cylindrical shells: the radius of a shell is the horizontal distance from \(x\) to the line \(x = \frac{\pi}{2}\), which is \(r = \frac{\pi}{2} - x\), and the height of the shell is \(h = \cos(x)\).
Write the volume integral as \(V = 2\pi \int_0^{\frac{\pi}{2}} (\text{radius})(\text{height}) \, dx = 2\pi \int_0^{\frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \cos(x) \, dx\).
Evaluate the integral by applying integration techniques such as integration by parts 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 a given 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 by revolving vertical or horizontal slices around an axis. It is especially useful when the axis of rotation is vertical and not the y-axis, as in this problem where the region is revolved around x = π/2.
Recommended video:
07:33Euler's Method
Trigonometric Functions and Their Graphs
Understanding the behavior of y = cos(x) on the interval [0, π/2] is essential. The cosine function decreases from 1 to 0 in this range, defining the boundary of the region to be revolved. This knowledge helps set up correct limits and integrand expressions for volume calculation.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
b. Show that the inner and outer surfaces of the solid have infinite area.
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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 (i) about the x-axis.
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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:
b. π ≤ x ≤ 2π.
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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 (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 3 of 1/√(x + 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 (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx