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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 6.3.17
Chapter 6, Problem 6.3.17

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=2x,y=0 , and x=3; about the x-axis (Verify that your answer agrees with the volume formula for a cone.)
Graph showing region bounded by y=2x, y=0, and x=3, shaded area labeled R, with axes and point (3,6).

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Identify the region R bounded by the curves y = 2x, y = 0, and x = 3. This region forms a right triangle with vertices at (0,0), (3,0), and (3,6).
Since the region is revolved about the x-axis, use the disk method to find the volume. The volume of the solid is given by integrating the area of circular cross-sections perpendicular to the x-axis.
The radius of each disk is the y-value of the curve y = 2x at a given x, so the radius r(x) = 2x. The area of each disk is A(x) = \( \pi \times (2x)^2 = 4\pi x^2 \).
Set up the integral for the volume V as \( V = \int_0^3 4\pi x^2 \, dx \), where the limits of integration are from x = 0 to x = 3.
Evaluate the integral to find the volume. After integration, compare your result with the volume formula for a cone, \( V = \frac{1}{3} \pi r^2 h \), where the radius r = 6 and height h = 3, to verify your answer.

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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 an axis. The volume can be computed using methods like the disk/washer or shell method, which integrate cross-sectional areas perpendicular to the axis of rotation.
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Finding Volume Using Disks

Disk Method

The disk method calculates volume by slicing the solid into thin disks perpendicular to the axis of rotation. Each disk's volume is approximated by π(radius)^2 times thickness, and integrating these volumes over the interval gives the total volume.
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Disk Method Using y-Axis

Volume Formula for a Cone

The volume of a cone is given by (1/3)πr^2h, where r is the base radius and h is the height. Verifying the volume of the solid of revolution against this formula confirms the correctness of the integral setup and solution.
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Finding Volume Using Disks
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