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.
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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 8.8.90c
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.
Verified step by step guidance1
First, identify the region bounded by the curves: \( y = \frac{1}{\sqrt{x}} \), \( y = 0 \), \( x = 0 \), and \( x = 1 \). This region lies in the first quadrant between \( x = 0 \) and \( x = 1 \), above the x-axis and below the curve \( y = \frac{1}{\sqrt{x}} \).
Since the solid is formed by revolving this region about the y-axis, we will use the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around the y-axis is: \(\n\[\n\)\[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \]\(\n\]\nwhere\) the radius is the distance from the y-axis (which is \( x \)) and the height is the function value \( y = \frac{1}{\sqrt{x}} \).
Set up the integral with the limits of integration from \( x = 0 \) to \( x = 1 \): \(\n\[\n\)\[ V = 2\pi \int_{0}^{1} x \cdot \frac{1}{\sqrt{x}} \, dx \]\(\n\]\nSimplify\) the integrand before integrating.
Simplify the integrand \( x \cdot \frac{1}{\sqrt{x}} = x \cdot x^{-1/2} = x^{1 - \frac{1}{2}} = x^{\frac{1}{2}} \). So the integral becomes \(\n\)\(\n\)\[ V = 2\pi \int_{0}^{1} x^{\frac{1}{2}} \, dx \]
Evaluate the integral \( \int_{0}^{1} x^{\frac{1}{2}} \, dx \) using the power rule for integration: \(\n\[\n\)\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]\(\n\]\nApply\) the limits from 0 to 1 and multiply by \( 2\pi \) to express the volume.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Setting up the region bounded by curves
Understanding the region involves identifying the area enclosed by the given curves y = 1/√x, y = 0, x = 0, and x = 1 in the first quadrant. This means recognizing the limits of integration and the shape formed, which is essential before applying volume formulas.
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Finding Area Between Curves on a Given Interval
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 equal to the x-value, height given by the function, and thickness dx, allowing calculation of the solid's volume.
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Integration with respect to x
Since the region is bounded between x = 0 and x = 1, and the function is given as y in terms of x, the volume integral is set up with respect to x. Proper integration techniques must be applied to evaluate the integral and find the exact volume.
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Related Practice
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?
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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 (ii) about the y-axis.
1
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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
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 2 to 4 of 1/(s - 1)² ds