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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 6.3.74a
Chapter 6, Problem 6.3.74a

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.

Verified step by step guidance
1
Identify the region R bounded by the curves \(y = x^{\frac{1}{n}}\) and \(y = x^n\) in the first quadrant. Since \(n > 1\), these two curves intersect at \(x=0\) and \(x=1\) because \(x^{\frac{1}{n}} = x^n\) implies \(x=0\) or \(x=1\).
Set up the volume integral using the method of washers (disks with holes) when revolving around the x-axis. The volume \(V(n)\) is given by integrating the difference of the squares of the outer and inner radii:
\[V(n) = \pi \int_0^1 \left[ (\text{outer radius})^2 - (\text{inner radius})^2 \right] \, dx.\]
Determine which curve is the outer radius and which is the inner radius with respect to the x-axis. Since \(y = x^{\frac{1}{n}}\) is greater than \(y = x^n\) on \((0,1)\) for \(n > 1\), the outer radius is \(x^{\frac{1}{n}}\) and the inner radius is \(x^n\).
Write the integral explicitly as: \[V(n) = \pi \int_0^1 \left[ (x^{\frac{1}{n}})^2 - (x^n)^2 \right] \, dx = \pi \int_0^1 \left( x^{\frac{2}{n}} - x^{2n} \right) \, dx.\] Then, integrate term-by-term using the power rule for integrals.

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Key Concepts

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

Defining the Region Bounded by Curves

Understanding the region R requires identifying the curves y = x^(1/n) and y = x^n and their intersection points in the first quadrant. Since n > 1, these functions define a closed area between them for x in [0,1]. Recognizing the limits of integration is essential for setting up the volume integral.
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Finding Area When Bounds Are Not Given

Volume of Solids of Revolution Using the Disk/Washer Method

When a region is revolved about the x-axis, the volume can be found by integrating the area of circular cross-sections perpendicular to the axis. The washer method applies when there is an inner and outer radius, calculated from the bounding curves, and the volume is the integral of π(outer radius² - inner radius²) dx over the interval.
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Finding Volume Using Disks

Integration of Power Functions

The volume integral involves integrating expressions like x^(2/n) and x^(2n), which are power functions. Applying the power rule for integration, ∫x^m dx = x^(m+1)/(m+1), is necessary to find a closed-form expression for V(n) in terms of n.
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Related Practice
Textbook Question

40–43. Population growth


A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.


a. What is the population after 20 days? After 40 days?

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Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days). 

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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.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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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

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Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

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