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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 6.4.28
Chapter 6, Problem 6.4.28

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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First, identify the region R bounded by the curves: \( y = \sqrt{\sin^{-1} x} \), \( y = \sqrt{\frac{\pi}{2}} \), and \( x = 0 \). Understand that \( \sin^{-1} x \) is the inverse sine function, also called arcsin, which outputs values in radians.
Since the solid is generated by revolving the region about the x-axis, and the shell method involves cylindrical shells formed by revolving vertical slices, consider using horizontal slices instead. However, the shell method is typically easier when revolving around the x-axis if we integrate with respect to \( y \).
Express \( x \) in terms of \( y \) from the equation \( y = \sqrt{\sin^{-1} x} \). Square both sides to get \( y^2 = \sin^{-1} x \), then solve for \( x \) as \( x = \sin(y^2) \). This will be the radius function for the shells.
Set up the volume integral using the shell method formula for revolution about the x-axis: \(\n\[\n\)\[ V = 2\pi \int_{a}^{b} (\text{radius}) \times (\text{height}) \, dy \]\(\n\]\nHere\), the radius is the distance from the x-axis to the shell, which is \( y \), and the height is the horizontal length of the shell, which is \( x = \sin(y^2) - 0 = \sin(y^2) \). The limits of integration \( a \) and \( b \) correspond to the y-values bounding the region, from \( y = 0 \) to \( y = \sqrt{\frac{\pi}{2}} \).
Write the integral explicitly: \(\n\[\n\)\[ V = 2\pi \int_{0}^{\sqrt{\frac{\pi}{2}}} y \cdot \sin(y^2) \, dy \]\(\n\]\nThis\) integral represents the volume of the solid generated by revolving the region about the x-axis using the shell method. The next step would be to evaluate this integral, possibly using substitution, but as per instructions, we stop here.

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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's volume is approximated by its circumference times height times thickness. This method is especially useful when the axis of rotation is parallel to the axis of the variable of integration.
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Finding Volume Using Disks

Region Bounded by Curves

Understanding the region bounded by the given curves is essential to set correct integration limits. Here, the region is bounded by y = √(sin⁻¹x), y = √(π/2), and x = 0, which defines the shape and size of the area to be revolved around the x-axis.
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Finding Area When Bounds Are Not Given

Revolution About the x-axis

Revolving a region about the x-axis means rotating it horizontally, which affects how the shells are formed. For the shell method, the radius of each shell is the distance from the shell to the axis of rotation, and the height corresponds to the function values along the axis perpendicular to the x-axis.
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Disk Method Using y-Axis
Related Practice
Textbook Question

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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Textbook Question

Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 1/5 kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W = ∫a^bF(y) dy, where F is the variable force required to lift an object along a vertical line from y=a to y=b.)

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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

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=1−x^3, the x-axis, and the y-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

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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