Skip to main content
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
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

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.
Recommended video:
07:33
Euler's Method

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.
Recommended video:
06:51
Derivatives Applied To Acceleration Example 2

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.
Recommended video:
04:48
Finding Volume Using Disks
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]?

57
views
Textbook Question

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]. 

45
views
Textbook Question

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=cos x, for 0≤x≤π/2; about the x-axis

62
views
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. "

67
views
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

49
views
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.

51
views