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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

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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

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.


y = (x−2)³ −2,x=0, and y=25; about the y-axis

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

Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = x^3/2 / 3 − x^1/2 on [4, 16]

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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

Textbook Question

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.


Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.


y=(3x)^1/3 , for 0≤x≤8/3; about the y-axis