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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 6.4.76b
Chapter 6, Problem 6.4.76b

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).


b. How is this formula changed if x0>b?

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Step 1: Understand the problem setup. The region R is bounded by two functions, y = f(x) and y = g(x), on the interval [a, b], where f(x) ≥ g(x). The formula for the volume of the solid of revolution depends on the axis of rotation and the interval of integration.
Step 2: Recall the formula for the volume of a solid of revolution. If the region is revolved around the x-axis, the volume is calculated using the formula: V = π ∫ab[f(x)2 - g(x)2] dx. This formula assumes the interval of integration is [a, b].
Step 3: Consider the case where x₀ > b. If the upper limit of integration changes to x₀, the interval of integration becomes [a, x₀]. The formula for the volume is updated accordingly: V = π ∫ax₀[f(x)2 - g(x)2] dx. This reflects the new upper limit of integration.
Step 4: Verify the conditions for the functions f(x) and g(x) over the interval [a, x₀]. Ensure that f(x) ≥ g(x) holds true for all x in [a, x₀]. If this condition is violated, the formula may need adjustment to account for the change in the relationship between the functions.
Step 5: Adjust the formula if the axis of rotation changes. If the region is revolved around a different axis (e.g., the y-axis or a line other than the x-axis), the formula for the volume will need to be modified to account for the new geometry. This involves using the appropriate radius of rotation and updating the integral accordingly.

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Key Concepts

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

Volume of Revolution

The volume of revolution refers to the volume generated when a region in the plane is rotated around a specified axis. This is typically calculated using methods such as the disk method or the washer method, which involve integrating the area of circular cross-sections perpendicular to the axis of rotation. Understanding this concept is crucial for solving problems related to the volume of regions bounded by functions.
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Finding Volume Using Disks

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points on the x-axis. In the context of volume of revolution, they help determine the total volume by integrating the area of cross-sections over the interval [a, b]. Mastery of definite integrals is essential for applying the formulas correctly and understanding how changes in the limits of integration affect the result.
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Definition of the Definite Integral

Axis of Revolution

The axis of revolution is the line around which a region is rotated to create a three-dimensional solid. The choice of axis (e.g., x-axis, y-axis, or a vertical line) significantly influences the volume calculation. When the axis of revolution is outside the bounds of the region, as indicated by x0 > b, the formula for volume must be adjusted to account for the new limits and the geometry of the resulting solid.
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Disk Method Using y-Axis
Related Practice
Textbook Question

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

b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.

Textbook Question

Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.


b. How far does the car travel in the time it takes to come to rest?

Textbook Question

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.

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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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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


b. If f is not one-to-one on the interval [a, b], then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.