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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.R.55b
Chapter 6, Problem 6.R.55b

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?

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First, identify the region bounded by the curves: \(y = 2x\), \(y = 6 - x\), and \(y = 0\). Sketching or analyzing these will help visualize the area to be revolved.
Determine the points of intersection between \(y = 2x\) and \(y = 6 - x\) by setting \(2x = 6 - x\) and solving for \(x\). This gives the limits of integration for the volume calculation.
For the solid revolved about the line \(y = -2\), use the disk/washer method because the axis of rotation is horizontal and outside the region. Express the radius of each disk/washer as the distance from the curve to the line \(y = -2\).
Set up the integral for the volume about \(y = -2\) by integrating with respect to \(x\) between the intersection points. The volume element is \(\pi \times (\text{outer radius}^2 - \text{inner radius}^2) \, dx\), where the radii are distances from the curves to \(y = -2\).
For the solid revolved about the line \(x = -2\), use the shell method because the axis of rotation is vertical and outside the region. Express the height of each shell as the difference between the two curves, and the radius as the distance from \(x\) to \(-2\). Set up the integral with respect to \(x\) using the formula \(2\pi \times (\text{radius}) \times (\text{height}) \, dx\) over the same interval.

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Key Concepts

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

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around a line (axis). The volume is computed by integrating cross-sectional areas perpendicular to the axis of rotation, using methods like disks, washers, or shells.
Recommended video:
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Finding Volume Using Disks

Disk/Washer Method

The disk/washer method calculates volume by slicing the solid perpendicular to the axis of rotation, forming circular cross-sections (disks) or rings (washers). The volume is found by integrating the area of these disks or washers along the axis.
Recommended video:
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Disk Method Using y-Axis

Shell Method

The shell method involves slicing the solid parallel to the axis of rotation, creating cylindrical shells. The volume is computed by integrating the lateral surface area of these shells, which is useful when the disk/washer method is complicated.
Recommended video:
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Euler's Method
Related Practice
Textbook Question

58–61. Arc length Find the length of the following curves.

y = 2x+4 on [−2,2] (Use calculus.)

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Textbook Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.

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Textbook Question

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.


Use the shell method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R₃ about the line x=3. Do not evaluate the integral.

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Textbook Question

Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high.


b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?

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Textbook Question

58–61. Arc length Find the length of the following curves.

y = x³/6 + 1/2x on [1,2]

Textbook Question

Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?

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