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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 6.4.49b
Chapter 6, Problem 6.4.49b

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


b. Repeat part (a) using the disk method.

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Identify the region R bounded by the upper half of the circle \(x^{2} + y^{2} = r^{2}\) and the x-axis. Since it is the upper half, the function describing the curve is \(y = \sqrt{r^{2} - x^{2}}\) for \(x\) in \([-r, r]\).
Set up the volume integral using the disk method. When revolving around the x-axis, the volume is given by \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the radius of the disk at position \(x\).
Substitute \(f(x) = \sqrt{r^{2} - x^{2}}\) into the formula, so the volume integral becomes \(V = \pi \int_{-r}^{r} (\sqrt{r^{2} - x^{2}})^{2} \, dx\).
Simplify the integrand: \((\sqrt{r^{2} - x^{2}})^{2} = r^{2} - x^{2}\), so the integral is \(V = \pi \int_{-r}^{r} (r^{2} - x^{2}) \, dx\).
Evaluate the definite integral \(\int_{-r}^{r} (r^{2} - x^{2}) \, dx\) by integrating term-by-term and then multiply the result by \(\pi\) to find the volume of the sphere.

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Key Concepts

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

Disk Method

The disk method is a technique for finding the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation. Each slice forms a disk whose volume is approximated by π(radius)²(thickness). Integrating these volumes over the interval gives the total volume.
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Disk Method Using y-Axis

Equation of a Circle and Region Definition

The region R is bounded by the upper half of the circle x² + y² = r² and the x-axis, meaning y = √(r² - x²) for x in [-r, r]. Understanding this curve is essential to set up the integral for the volume calculation.
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Parameterizing Equations of Circles & Ellipses

Volume of Revolution about the x-axis

Revolving a region around the x-axis generates a 3D solid. The volume is found by integrating the cross-sectional areas (disks) perpendicular to the x-axis, where each radius corresponds to the y-value of the function defining the region.
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Finding Volume Using Disks
Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?

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Textbook Question

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.

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Textbook Question

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=tan x , for 0≤x≤π/4; about the x-axis 

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Textbook Question

Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.


b. How far does the probe fall in the first 30 s after it is released?

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Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

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Textbook Question

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.


b. Use the washer method to write an integral for the volume of the torus.

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