Textbook Question
Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.
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9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
1:29 minutesProblem 6.4.8a
Textbook Question
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and 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 guidance1
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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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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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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