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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.42a
Chapter 6, Problem 6.R.42a

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.

Verified step by step guidance
1
First, identify the region R bounded by the parabola \(y = 4 - x^{2}\) and the coordinate axes in the first quadrant. Since we are in the first quadrant, \(x \geq 0\) and \(y \geq 0\).
Express \(x\) as a function of \(y\) to set up the integral with respect to \(y\). Starting from \(y = 4 - x^{2}\), solve for \(x\): \(x = \sqrt{4 - y}\).
Since the solid is formed by revolving the region around the y-axis, the cross-sectional disks will be horizontal slices perpendicular to the y-axis. The radius of each disk is the \(x\)-value at that \(y\), which is \(r(y) = \sqrt{4 - y}\).
The area of each disk is \(A(y) = \pi [r(y)]^{2} = \pi (4 - y)\). The volume is found by integrating these areas along the \(y\)-axis from the lowest to the highest \(y\)-value in the region, which are \(y=0\) to \(y=4\).
Set up the volume integral using the disk method: \(V = \int_{0}^{4} \pi (4 - y) \, d y\). This integral will give the volume of the solid when evaluated.

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Key Concepts

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

Disk Method for Volume

The disk method calculates the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation into thin disks. Each disk's volume is approximated by π(radius)²(thickness), and integrating these volumes over the given interval yields the total volume.
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Finding Volume Using Disks

Rewriting Functions in Terms of y

When integrating with respect to y, it is necessary to express the function x in terms of y. This involves solving the given equation y = 4 - x² for x, which allows setting up the integral with y as the variable of integration.
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Completing the Square to Rewrite the Integrand

Bounds of Integration in the First Quadrant

The region is bounded by the parabola and coordinate axes in the first quadrant, so the limits of integration correspond to the y-values where the region exists, typically from y = 0 up to the maximum y-value on the parabola within the first quadrant.
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Finding Area When Bounds Are Not Given
Related Practice
Textbook Question

14–25. {Use of Tech} Areas of regions Determine the area of the given region.


The region in the first quadrant bounded by y = x/6 and y = 1−|x/2−1|

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.

Find the area of each of the regions R₁,R₂, and R₃.

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 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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Textbook Question

Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.


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

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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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.