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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 6.3.55
Chapter 6, Problem 6.3.55

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.


y=2 sin x and y=0 on [0,π]; about y=−2

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Identify the region R bounded by the curves y = 2 \(\sin\) x and y = 0 on the interval [0, \(\pi\)]. This region lies between the curve and the x-axis from x = 0 to x = \(\pi\).
Since the region is revolved about the line y = -2, which is horizontal and below the x-axis, use the washer method to find the volume of the solid of revolution.
Set up the radius expressions for the washers: the outer radius R(x) is the distance from y = -2 to y = 2 \(\sin\) x, which is R(x) = (2 \(\sin\) x) - (-2) = 2 \(\sin\) x + 2; the inner radius r(x) is the distance from y = -2 to y = 0, which is r(x) = 0 - (-2) = 2.
Write the volume integral using the washer formula: \(V = \pi \int_0^{\pi} \left[ R(x)^2 - r(x)^2 \right] \, dx = \pi \int_0^{\pi} \left[ (2 \sin x + 2)^2 - 2^2 \right] \, dx\).
Expand the integrand, simplify the expression inside the integral, and then integrate term-by-term with respect to x over [0, \(\pi\)] 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 a line. The volume can be computed using methods like the disk/washer or shell method, depending on the axis of rotation and the shape of the region.
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Finding Volume Using Disks

Washer Method

The washer method calculates volume by slicing the solid perpendicular to the axis of rotation, forming washers (disks with holes). The volume is found by integrating the difference of the outer and inner radii squared, multiplied by π, over the given interval.
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Euler's Method

Adjusting for Axis of Rotation Not on the Coordinate Axis

When the axis of rotation is not the x- or y-axis (here y = -2), distances (radii) must be measured relative to this line. This requires shifting the function values accordingly to find the correct radii for the washers or shells.
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Disk Method Using y-Axis
Related Practice
Textbook Question

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

x = y⁴/4 + 1/8y², for 1≤y≤2

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Textbook Question

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


y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis 

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Textbook Question

Explain how to find the mass of a one-dimensional object with a variable density ρ.

Textbook Question

For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.


R is bounded by y=4−2x, the x-axis, and the y-axis.

Textbook Question

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


y=4x−1, for 1≤x≤4; about the y-axis (Hint: Integrate with respect to y.)

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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.

y = (1+x²)^−1,y = 0,x = 0, and x = 2; about the y-axis