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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.3.6b
Chapter 6, Problem 6.3.6b

Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures). 


b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2]. 

Verified step by step guidance
1
Identify the region R bounded by the curve \(y = \cos^{-1} x\) and the x-axis on the interval \([0,1]\). Since \(y = \cos^{-1} x\), we can rewrite this as \(x = \cos y\) for \(y\) in \([0, \frac{\pi}{2}]\) because \(\cos^{-1} x\) maps \([0,1]\) to \([0, \frac{\pi}{2}]\).
Understand that the solid is formed by revolving the region R about the y-axis. For a fixed \(y\) in \([0, \frac{\pi}{2}]\), the cross section perpendicular to the y-axis is a circle.
Determine the radius of the cross-sectional circle at height \(y\). Since the solid is revolved around the y-axis, the radius is the horizontal distance from the y-axis to the curve, which is \(x = \cos y\).
Write the formula for the area \(A(y)\) of the cross section at height \(y\). The area of a circle is \(\pi\) times the radius squared, so \(A(y) = \pi (\cos y)^2\).
Express the final formula for the cross-sectional area as \(A(y) = \pi \cos^2 y\) for \(y\) in \([0, \frac{\pi}{2}]\).

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Key Concepts

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

Inverse Trigonometric Functions

The function y = cos⁻¹(x) is the inverse cosine function, which maps values from [0,1] to angles in [0, π/2]. Understanding its domain and range is essential to relate x and y coordinates and to express x as a function of y when analyzing the region and cross sections.
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Derivatives of Other Inverse Trigonometric Functions

Cross-Sectional Area of a Solid of Revolution

A cross section perpendicular to the axis of revolution is a shape whose area depends on the distance from the axis. For revolution about the y-axis, the cross section at height y is typically a disk or washer, and its area A(y) can be found using the radius determined by the x-values corresponding to y.
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Introduction to Cross Sections

Relationship Between x and y in the Region

Since y = cos⁻¹(x), we can rewrite x in terms of y as x = cos(y). This relationship allows us to express the radius of the cross section at height y, which is crucial for setting up the area formula A(y) for the solid's cross section.
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Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?

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Textbook Question

Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

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Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

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Textbook Question

"Determine whether the following statements are true and give an explanation or counterexample.


b. The volume of a hemisphere can be computed using the disk method. "

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Textbook Question

21–30. {Use of Tech} Arc length by calculator


b. If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π

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Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.

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