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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 6.4.49a
Chapter 6, Problem 6.4.49a

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.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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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. This means \(y\) ranges from 0 to \(\sqrt{r^{2} - x^{2}}\) for \(x\) in \([-r, r]\).
Since we are revolving the region R about the x-axis, set up the shell method using vertical shells. The shells will be formed by slicing the region horizontally at a height \(y\), where \(y\) ranges from 0 to \(r\).
For the shell method, the radius of a shell is the distance from the x-axis to the shell, which is simply \(y\). The height of the shell is the length of the horizontal segment inside the circle at height \(y\), which can be found by solving for \(x\): \(x = \pm \sqrt{r^{2} - y^{2}}\). So the height is \(2 \sqrt{r^{2} - y^{2}}\).
The volume of each cylindrical shell is given by the formula \(dV = 2 \pi \times (\text{radius}) \times (\text{height}) \times (\text{thickness})\). Here, thickness is \(dy\), so \(dV = 2 \pi y \cdot 2 \sqrt{r^{2} - y^{2}} \ dy = 4 \pi y \sqrt{r^{2} - y^{2}} \ dy\).
Integrate the volume of the shells from \(y=0\) to \(y=r\) to find the total volume: \(V = \int_{0}^{r} 4 \pi y \sqrt{r^{2} - y^{2}} \ dy\). Evaluate this integral (using substitution if needed) to verify that the volume equals \(\frac{4}{3} \pi r^{3}\).

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Key Concepts

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

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell has a radius, height, and thickness, and the volume is found by summing these shells along the axis of revolution. It is especially useful when revolving regions around an axis parallel to the axis of the function.
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Finding Volume Using Disks

Equation of a Circle and Region Definition

The region R is defined by the upper half of the circle x² + y² = r² and the x-axis, meaning y ≥ 0. Understanding this boundary helps set the limits of integration and the height of the shells when revolving the region around the x-axis.
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Guided course
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Parameterizing Equations of Circles & Ellipses

Volume Formula of a Sphere

The volume of a sphere with radius r is given by (4/3)πr³. This formula can be derived using integral calculus methods like the shell method or disk/washer method by revolving a semicircular region around an axis.
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Finding Volume Using Disks
Related Practice
Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days). 

Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = 1/x, for 1 ≤ x ≤ 10

Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = −t³+3t²−2t on [0, 3]; s(0)=4

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.


a. Use the shell method to write an integral for the volume of the torus.

Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).