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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 6.4.5b
Chapter 6, Problem 6.4.5b

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]?

Verified step by step guidance
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Step 1: Understand the shell method. The shell method involves calculating the volume of a solid of revolution by integrating the product of the shell radius, shell height, and 2π over the interval of interest.
Step 2: Identify the shell height. The shell height is the vertical distance between the upper curve y=2−x² and the lower curve y=x at a given x-value. This is calculated as the difference between the two functions: height = (2−x²)−x.
Step 3: Simplify the expression for the shell height. Combine like terms to express the height as a single function: height = 2−x²−x.
Step 4: Confirm the interval of integration. The region R is bounded in the first quadrant, and the intersection points of the curves y=2−x² and y=x occur at x=0 and x=2. Thus, the interval of integration is [0, 2].
Step 5: Use the shell height expression in the shell method formula. The shell height at a point x in [0, 2] is given by the simplified expression height = 2−x²−x, which will be used in the integral to compute the volume.

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Key Concepts

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

Cylindrical Shell Method

The cylindrical shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells, where the volume of each shell is calculated and then integrated over the interval of interest. This method is particularly useful when revolving around the y-axis, as it simplifies the integration process by using the height and radius of the shells.
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Height of the Shell

In the context of the shell method, the height of a cylindrical shell at a point x is determined by the difference between the upper and lower bounding functions of the region being revolved. For the given problem, the height is calculated as the vertical distance between the curve y = 2 - x² and the line y = x, which represents the top and bottom of the region R, respectively.
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Volume of Revolution

The volume of revolution refers to the volume of a three-dimensional solid formed by rotating a two-dimensional shape around an axis. The volume can be calculated using various methods, including the disk, washer, and shell methods. Understanding how to set up the integral for volume is crucial, as it involves determining the appropriate bounds and the function that describes the shape being revolved.
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Related Practice
Textbook Question

Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.

b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

Textbook Question

Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.


b. When, if ever, does Alicia overtake Boris?

Textbook Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


b. Find the function that gives the amount of water in the tank at any time t≥0.

Textbook Question

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.


b. Use the washer method to write an integral for the volume of the torus.

Textbook Question

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

b. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

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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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