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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 6.4.8a
Chapter 6, Problem 6.4.8a

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Graph showing region R bounded by curves y=2, y=2−√x, and line x=4 in the first quadrant.


Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.


a. What is the radius of a cylindrical shell at a point x in [0, 4]?

Verified step by step guidance
1
Identify the axis of rotation: The region R is revolved about the vertical line x=4. This means the radius of a cylindrical shell at a point x is the horizontal distance from x to the line x=4.
Determine the formula for the radius: The radius is given by the difference between the x-coordinate of the line of rotation (x=4) and the x-coordinate of the shell. Thus, the radius is r(x) = 4 - x.
Verify the bounds of integration: The region R is bounded by x-values from x=0 to x=4, so the radius formula r(x) = 4 - x is valid for all x in [0, 4].
Understand the shell method setup: In the shell method, the radius r(x) is multiplied by the height of the shell and integrated over the bounds of x to compute the volume. For this part of the problem, we are only determining the radius.
Conclude the radius: The radius of a cylindrical shell at a point x in [0, 4] is r(x) = 4 - x.

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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 a region around a vertical line, as it simplifies the calculation of the radius and height of each shell.
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Euler's Method

Radius of a Shell

In the context of the shell method, the radius of a cylindrical shell is the distance from the axis of rotation to the shell itself. For the given problem, where the region R is revolved around the line x=4, the radius at a point x is determined by the difference between the line x=4 and the x-coordinate of the shell, which is expressed as (4 - x). This distance is crucial for calculating the volume of the shell.
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Radius of Convergence

Bounded Region

The bounded region R in the problem is defined by the curves y = 2, y = 2 - √x, and the vertical line x = 4. Understanding the boundaries of this region is essential for setting up the integral for volume calculation. The area enclosed by these curves in the first quadrant provides the shape that will be revolved around the line x=4 to generate the solid.
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Area of Polar Regions
Related Practice
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.

a. How much work is required to wind the entire chain onto the cylinder using the winch?

Textbook Question

Find the area of the region (see figure) in two ways.

a. Using integration with respect to x.

Textbook Question

Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).

a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank?

Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


a. Verify that the amount of blood pumped over a one-second interval is 70 mL.

Textbook Question

Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.


a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.

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Textbook Question

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


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = x³/3, for −1≤x≤1