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Ch. 6 - Applications of Definite Integrals
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 6 - Applications of Definite IntegralsProblem 6.PE.12
Chapter 6, Problem 6.PE.12

Volumes
Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

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Identify the region bounded by the curve \(y = \sin x\), the vertical lines \(x = 0\) and \(x = \pi\), and the horizontal line \(y = 2\). This region lies between \(x=0\) and \(x=\pi\) and between \(y=\sin x\) and \(y=2\).
Since the solid is generated by revolving this region about the line \(y = 2\), use the method of washers (disks with holes). The axis of rotation is horizontal and above the curve \(y = \sin x\).
Set up the volume integral using the washer method. The outer radius \(R\) is the distance from the line \(y=2\) to the lower boundary \(y=\sin x\), and the inner radius \(r\) is the distance from \(y=2\) to the line \(y=2\) itself (which is zero). So, \(R = 2 - \sin x\) and \(r = 0\).
Write the volume integral as \(V = \pi \int_0^{\pi} \left(R^2 - r^2\right) \, dx = \pi \int_0^{\pi} (2 - \sin x)^2 \, dx\).
Expand the integrand \((2 - \sin x)^2\), simplify, and then integrate term-by-term 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 given axis. The volume is typically calculated using integral calculus methods such as the disk, washer, or shell methods, depending on the axis and shape of the region.
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Finding Volume Using Disks

Washer Method

The washer method is used when the solid has a hollow center, formed by revolving a region between two curves around an axis. The volume is found by integrating the difference of the squares of the outer and inner radii, representing the outer and inner disks (washers) perpendicular to the axis of rotation.
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Euler's Method

Setting up the Integral with Respect to the Axis of Rotation

To correctly compute the volume, one must express the radii of the washers in terms of the distance from the axis of rotation. When revolving around a line other than the x- or y-axis, adjust the radius formulas accordingly, such as using (2 - y) or (y - 2) if revolving around y = 2.
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Integration by Parts for Definite Integrals Example 8
Related Practice
Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

c. the line x = 5

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

d. the line y = 2

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Textbook Question

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_______

y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

a. the x-axis

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about

c. the line x = 1

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Textbook Question

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.

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