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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.66c
Chapter 6, Problem 6.R.66c

Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].


c. Find the volume of the solid generated when R is revolved about the x-axis.

Verified step by step guidance
1
Identify the region R bounded by the curve \(f(x) = \frac{1}{3}x^{3}\) and the x-axis on the interval \([0, 2]\). Since \(f(x) \geq 0\) on this interval, the region lies above the x-axis.
Recall the formula for the volume of a solid of revolution generated by revolving a region around the x-axis: \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\).
Substitute the given function and interval into the formula: \(V = \pi \int_{0}^{2} \left( \frac{1}{3}x^{3} \right)^{2} \, dx\).
Simplify the integrand: \(\left( \frac{1}{3}x^{3} \right)^{2} = \frac{1}{9} x^{6}\), so the integral becomes \(V = \pi \int_{0}^{2} \frac{1}{9} x^{6} \, dx\).
Set up the integral for evaluation: \(V = \frac{\pi}{9} \int_{0}^{2} x^{6} \, dx\). The next step would be to compute this integral and then multiply by \(\frac{\pi}{9}\) 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 an axis. The volume can be computed using integral calculus, often applying the disk or washer method, which sums up infinitesimal circular cross-sections 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 times the square of the radius (distance from the axis to the function) times the thickness (dx). Integrating these volumes over the interval gives the total volume.
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Disk Method Using y-Axis

Definite Integration

Definite integration is used to sum continuous quantities over an interval. In this problem, it allows the calculation of the total volume by integrating the area of cross-sectional disks from x = 0 to x = 2, providing an exact value for the volume of the solid.
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Definition of the Definite Integral
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