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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 6.3.24
Chapter 6, Problem 6.3.24

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x and y=4√x; about the x-axis
Graph showing region bounded by y = x and y = 4√x, shaded area revolved around x-axis forming a solid.

Verified step by step guidance
1
Identify the region R bounded by the curves \(y = x\) and \(y = 4\sqrt{x}\), and the axis of revolution, which is the x-axis.
Find the points of intersection of the curves by setting \(x = 4\sqrt{x}\). Solve for \(x\) to determine the limits of integration.
Since the region is revolved around the x-axis, use the washer method to find the volume. The volume element is given by \(\pi \int_a^b \left(R(x)^2 - r(x)^2\right) \, dx\), where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius.
Determine which curve is the outer radius and which is the inner radius relative to the x-axis. Here, the outer radius is the curve farther from the x-axis, and the inner radius is the closer curve. Express these radii as functions of \(x\).
Set up the integral for the volume using the limits of integration found in step 2 and the radii from step 4. The integral will be \(V = \pi \int_a^b \left( (4\sqrt{x})^2 - (x)^2 \right) \, dx\). Evaluate this integral to find the volume.

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Key Concepts

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

Finding the Region of Integration

To find the volume of the solid, first identify the region bounded by the curves y = x and y = 4√x. Determine their points of intersection by solving x = 4√x, which sets the limits of integration along the x-axis. This step is crucial to correctly define the interval over which the volume will be calculated.
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Volume of Solids of Revolution Using the Washer Method

When a region is revolved around the x-axis, the volume can be found using the washer method. This involves integrating the difference of the squares of the outer and inner radii (functions of x) multiplied by π over the interval. Here, the outer radius is the upper curve and the inner radius is the lower curve relative to the axis of rotation.
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Setting up and Evaluating the Definite Integral

After determining the limits and radii, set up the definite integral for volume: V = π∫[a to b] (R(x))^2 - (r(x))^2 dx. Carefully substitute the expressions for R(x) and r(x) from the given curves, then evaluate the integral using appropriate techniques such as substitution or power rule to find the exact volume.
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Related Practice
Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x−x⁴,y=0; about the x-axis.

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Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)

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Textbook Question

Filling a spherical tank A spherical water tank with an inner radius of 8 m has its lowest point 2 m above the ground. It is filled by a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how much work is required to fill the tank if it is initially empty?

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Textbook Question

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).


a(t) = cos2t; v(0) = 5; s(0) = 7

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Textbook Question

Find the area of the region described in the following exercises.


The region bounded by y=|x−3|and y=x/2

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Textbook Question

What is the area of the curved surface of a right circular cone of radius 3 and height 4?

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