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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 6.3.34
Chapter 6, Problem 6.3.34

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


y=√sin x,y=1, and x=0; about the x-axis 

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First, identify the region R bounded by the curves: \( y = \sqrt{\sin x} \), \( y = 1 \), and \( x = 0 \). Determine the interval for \( x \) where these curves intersect and define the region. Since \( y = \sqrt{\sin x} \) is defined where \( \sin x \geq 0 \), consider \( x \) in \( [0, \pi] \) because \( \sin x \) is positive there, and find the point where \( \sqrt{\sin x} = 1 \) to find the upper bound of \( x \).
Next, sketch or visualize the region and the axis of rotation, which is the x-axis. Since the region is bounded above by \( y = 1 \) and below by \( y = \sqrt{\sin x} \), revolving around the x-axis will create a solid with a hole (washer) at each cross-section.
Set up the volume integral using the washer method. The volume \( V \) is given by integrating the area of the washers perpendicular to the x-axis over the interval found in step 1. The outer radius \( R_{outer} \) is the distance from the x-axis to \( y = 1 \), and the inner radius \( R_{inner} \) is the distance from the x-axis to \( y = \sqrt{\sin x} \). So, the volume integral is: \[ V = \pi \int_{0}^{a} \left( R_{outer}^2 - R_{inner}^2 \right) \, dx = \pi \int_{0}^{a} \left(1^2 - (\sqrt{\sin x})^2 \right) \, dx \] where \( a \) is the upper limit of integration found in step 1.
Simplify the integrand inside the integral. Since \( (\sqrt{\sin x})^2 = \sin x \), the integrand becomes \( 1 - \sin x \). So the volume integral simplifies to: \[ V = \pi \int_{0}^{a} (1 - \sin x) \, dx \]
Finally, evaluate the integral (without calculating the numerical value here). Use the antiderivative of \( 1 - \sin x \), which is \( x + \cos x \), and apply the limits from 0 to \( a \). This will give the expression for the volume of the solid generated by revolving the region about the x-axis.

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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 an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Understanding how to set up these integrals is essential for solving volume problems.
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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, creating washers (disks with holes). The volume is found by integrating the difference between the outer and inner radii squared, multiplied by π, over the interval. This method is useful when the region is bounded by two curves.
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Euler's Method

Setting up Integration Limits and Radii

Correctly identifying the limits of integration and the radii of washers or shells is crucial. The limits come from the intersection points of the bounding curves, and the radii depend on the distance from the axis of rotation to the curves. Accurate setup ensures the integral represents the volume correctly.
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Improper Integrals: Infinite Intervals
Related Practice
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. 


{Use of Tech} y² = ln x,y² = ln x³, and y=2; about the x-axis

Textbook Question

Find the area of the region described in the following exercises.


The region bounded by x=y(y−1) and y=x/3

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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. 


{Use of Tech} y = √sin^−1x,y = √π/2, and x=0; about the x-axis

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Textbook Question

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

x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

Textbook Question

The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.

Textbook Question

Find the volume of the torus formed when the circle of radius 2 centered at (3, 0) is revolved about the y-axis. Use geometry to evaluate the integral.

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