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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 6.3.38
Chapter 6, Problem 6.3.38

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=4−x^2,x=2, and y=4; about the y-axis

Verified step by step guidance
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First, identify the region R bounded by the curves: \(y = 4 - x^2\), the vertical line \(x = 2\), and the horizontal line \(y = 4\). Sketching these curves helps visualize the region and the axis of rotation (the y-axis).
Since the solid is generated by revolving the region around the y-axis, consider using the method of cylindrical shells. The shell radius will be the distance from the y-axis, which is \(x\), and the shell height will be the vertical distance between the curves, which is \(y = 4 - x^2\).
Set up the volume integral using the shell method formula: \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\). Here, the radius is \(x\), the height is \(4 - x^2\), and the limits of integration are from \(x=0\) to \(x=2\) (since the region is bounded by \(x=2\) and the curve intersects the y-axis at \(x=0\)).
Write the integral explicitly: \(V = 2\pi \int_{0}^{2} x (4 - x^2) \, dx\). This integral represents the volume of the solid formed by revolving the region around the y-axis.
Finally, evaluate the integral by expanding the integrand and integrating term-by-term, then multiply by \(2\pi\) to find the volume. (Do not compute the final numerical value here, just set up the integral.)

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Key Concepts

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

Setting up the region bounded by curves

Understanding the region R requires identifying the area enclosed by the curves y = 4 - x², the vertical line x = 2, and the horizontal line y = 4. This involves sketching or analyzing the intersection points to determine the limits of integration and the shape of the region.
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Finding Area Between Curves on a Given Interval

Volume of solids of revolution about the y-axis

When a region is revolved about the y-axis, the volume of the resulting solid can be found using methods like the shell method or the disk/washer method. Choosing the appropriate method depends on the axis of rotation and the given curves.
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Disk Method Using y-Axis

Using the shell method for volume calculation

The shell method involves integrating cylindrical shells formed by revolving vertical slices of the region around the y-axis. The volume is computed by integrating 2π(radius)(height) with respect to x over the interval defining the region.
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Finding Volume Using Disks
Related Practice
Textbook Question

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.


v(t)=2 sin t, for 0≤t≤π

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Textbook Question

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


The region bounded by y=x^2−2x+1 and y=5x−9

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Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


x = x³ ,y = 1, and x = 0; about the x-axis

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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) = −32; v(0)=50; s(0)=0

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Textbook Question

Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.

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Textbook Question

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


The region bounded by y=√x, y=2x−15, and y=0

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