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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.52
Chapter 6, Problem 6.R.52

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 graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?

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First, visualize the region bounded by the curve \(y = 4 - x^{2}\) and the x-axis on the interval \([-2, 2]\). This region lies above the x-axis and below the parabola between \(x = -2\) and \(x = 2\).
Since the solid is generated by revolving this region about the vertical line \(x = -2\), consider using the shell method because the axis of rotation is vertical and the slices perpendicular to the x-axis will be easier to handle.
Set up a representative cylindrical shell at a position \(x\) between \(-2\) and \(2\). The height of the shell is given by the function value \(h(x) = 4 - x^{2}\), and the radius of the shell is the horizontal distance from \(x\) to the line \(x = -2\), which is \(r(x) = x - (-2) = x + 2\).
The volume of each thin shell is given by the formula \(dV = 2\pi \times \text{radius} \times \text{height} \times \text{thickness} = 2\pi (x + 2)(4 - x^{2}) \, dx\).
To find the total volume, integrate the expression for \(dV\) over the interval \(x = -2\) to \(x = 2\): \(V = \int_{-2}^{2} 2\pi (x + 2)(4 - x^{2}) \, dx\). This integral can then be evaluated 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.

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 of revolution). The volume is computed by integrating cross-sectional areas perpendicular to the axis, using methods like disks, washers, or shells depending on the axis and shape.
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Finding Volume Using Disks

Shell Method

The shell method calculates volume by summing cylindrical shells formed by revolving vertical or horizontal slices around an axis. Each shell's volume is 2π(radius)(height)(thickness), making it useful when the axis of rotation is parallel to the slices and not the coordinate axis.
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Euler's Method

Setting up the Integral with Correct Limits and Radius

Accurately determining the radius and height of slices relative to the axis of rotation is crucial. For rotation about x = -2, the radius is the horizontal distance from x to -2, and the height is given by the function y = 4 - x². Proper limits correspond to the interval of the region.
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Improper Integrals: Infinite Intervals
Related Practice
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a. Graph the velocity function, for t≥0.

Textbook Question

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).

a. Write a single integral that gives the area of R.

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c. Find a general expression for Vₓ in terms of a and p. Note that p=1/2 is a special case. What is Vₓ when p=1/2?

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Textbook Question

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:


a. Apply the disk method and integrate with respect to y.

Textbook Question

35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of

Find the area of the region R.