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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.13d
Chapter 6, Problem 6.PE.13d

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

Verified step by step guidance
1
First, identify the region bounded by the curve \(y = x^{2} - 2x\) and the x-axis. To do this, find the points where the curve intersects the x-axis by solving \(x^{2} - 2x = 0\).
Determine the interval of integration by finding the roots from the previous step. These roots will serve as the limits for the volume integral.
Since the solid is generated by revolving the region about the line \(y = 2\), calculate the radius of the washers formed. The outer radius \(R(x)\) is the distance from \(y=2\) to the x-axis (which is 0), and the inner radius \(r(x)\) is the distance from \(y=2\) to the curve \(y = x^{2} - 2x\). Express these radii as functions of \(x\).
Set up the volume integral using the washer method formula: \(V = \pi \int_{a}^{b} \left[R(x)^{2} - r(x)^{2}\right] \, dx\), where \(a\) and \(b\) are the limits found earlier.
Evaluate the integral to find the volume of the solid. Remember, the integral represents the sum of the volumes of infinitesimally thin washers formed by revolving the region around \(y=2\).

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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 is typically calculated using integral methods such as the disk, washer, or shell methods, depending on the axis of rotation and the shape of the region.
Recommended video:
04:48
Finding Volume Using Disks

Washer Method

The washer method is used when the solid has a hollow center, created by revolving a region around a line that does not coincide with the axis of the curve. It involves subtracting the volume of the inner radius from the outer radius, integrating π(outer radius² - inner radius²) dx or dy.
Recommended video:
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Euler's Method

Adjusting for Axis of Rotation (Line y = 2)

When revolving around a line other than the x- or y-axis, distances (radii) must be measured relative to that line. For y = 2, the radius is the vertical distance between the curve and y = 2, calculated as |2 - y|, which affects the limits and integrand in the volume integral.
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Disk Method Using y-Axis
Related Practice
Textbook Question

Volumes

Find the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4/x³ and the lines x = 1 and y = 1/2 about

c. the line x = 2

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded by the parabola y² = 4x and the line y = x about

b. the y-axis

1
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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

c. the line x = 5

1
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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 by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

1
views
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

1
views