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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

Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)


b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy

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Textbook Question

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


b. What is the SAV ratio of a ball with radius a? 

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 y-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 y in [1, 3]?

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Textbook Question

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


b. Repeat part (a) using the disk method.

Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

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