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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 6.3.22
Chapter 6, Problem 6.3.22

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=0,y=lnx,y=2, and x=0; about the y-axis

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First, identify the region R bounded by the curves: \(y=0\), \(y=\ln x\), \(y=2\), and \(x=0\). Sketching or visualizing these curves helps understand the shape and limits of the region.
Since the solid is generated by revolving the region around the y-axis, consider using the method of cylindrical shells or washers. Here, cylindrical shells are often convenient when revolving around the y-axis.
Express the boundaries in terms of \(x\) and \(y\). Note that \(y=\ln x\) can be rewritten as \(x = e^y\). The vertical boundaries are \(x=0\) and the curve \(x = e^y\), and the horizontal boundaries are \(y=0\) and \(y=2\).
Set up the volume integral using the shell method. The formula for the volume of a solid of revolution about the y-axis using shells is: \[V = \int_{a}^{b} 2\pi (\text{radius})(\text{height}) \, dy\] Here, the radius is the distance from the y-axis to the shell (which is \(x\)), and the height is the vertical thickness between \(y=0\) and \(y=\ln x\) or vice versa. Since we are integrating with respect to \(y\), the radius is \(e^y\) and the height is the difference between the outer and inner \(x\) values.
Determine the limits of integration for \(y\), which are from \(y=0\) to \(y=2\). Then write the integral explicitly as: \[V = \int_{0}^{2} 2\pi (e^y)(\text{height in terms of } y) \, dy\] Since the region is bounded by \(x=0\) and \(x=e^y\), the height corresponds to the horizontal distance between these two, which is \(e^y - 0 = e^y\). So the integrand becomes \(2\pi e^y \cdot e^y = 2\pi e^{2y}\). Finally, set up the integral and prepare to evaluate it.

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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 area enclosed by the given curves y=0, y=ln(x), y=2, and x=0. This involves interpreting the inequalities these curves impose on x and y, and visualizing or sketching the region to determine its shape and boundaries.
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Finding Area When Bounds Are Not Given

Volume of Solids of Revolution

When a region is revolved about an axis, it generates a 3D solid. Calculating its volume often uses methods like the disk/washer or shell method, which integrate cross-sectional areas or cylindrical shells perpendicular or parallel to the axis of revolution.
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Finding Volume Using Disks

Shell Method for Revolution About the y-Axis

The shell method is ideal for revolving regions around the y-axis. It involves integrating cylindrical shells with radius equal to the x-value, height given by the function difference, and thickness dx. The volume is found by integrating 2π(radius)(height) dx over the interval.
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Disk Method Using y-Axis
Related Practice
Textbook Question

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


The region bounded by y=e^x, y=2e^−x+1, and x=0

Textbook Question

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters. 


The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis 

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Textbook Question

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


The region bounded by y=2−|x|and y=x^2

Textbook Question

For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.


R is bounded by y=x^2 and y=√8x.

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Textbook Question

9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)


The work required to empty the tank through an outflow pipe at the top of the tank

Textbook Question

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=2−x^2; about the x-axis