method of cylindrical shells

a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable

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Calculus Volume 1 · 6.3 Volumes of Revolution: Cylindrical Shells

Volumes of Revolution: Cylindrical Shells: practice problems

25 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 115
For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.
[T] Bounded by the curves $y = 3 x, y = 0, and x = 3$ rotated around the $y -axis .$
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Exercises 117
For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.
[T] Bounded by the curves $y = 3 x, y = 0, and x = 3$ rotated around the $x -axis .$
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Exercises 119
For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.
[T] Bounded by the curves $y = 2 x^{3}, y = 0, and x = 2$ rotated around the $x -axis .$
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Exercises 121
For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$
$y = 5 x^{3}, x = 0, and x = 1$
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Exercises 123
For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$
$y = 1 - x^{2}, x = 0, and x = 1$
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Exercises 125
For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$
$y = sin x^{2}, x = 0, and x = π$
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Exercises 127
For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$
$y = x, x = 0, and x = 1$
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Exercises 129
For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$
$y = 5 x^{3} - 2 x^{4}, x = 0, and x = 2$
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Exercises 131
For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$
$y = x^{2}, x = 0, x = 2$ and the x-axis
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Exercises 133
For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$
$y = \frac{2}{x ^{2}}, x = 1, x = 2,$ and the -axis
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Exercises 135
For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$
$x = \frac{1 + y ^{2}}{y}, y = 1, y = 4,$ and the -axis
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Exercises 137
For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$
$x = y^{3} - 2 y^{2}, x = 0, x = 9$
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Exercises 139
For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$
$x = 3 27 y and x = \frac{3 y}{4}$
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Exercises 141
For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.
$y = x^{3}, x = 0, and y = 8$ rotated around the $y -axis .$
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Exercises 143
For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.
$y = x, y = 0, and x = 1$ rotated around the line $x = 2.$
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Exercises 145
For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.
$y = x and y = x^{2}$ rotated around the $y -axis .$
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Exercises 147
For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.
$x = y^{3}, x = \frac{1}{y}, x = 1, and x = 2$ rotated around the
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Exercises 149
For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.
[T] Left of $x = sin (π y),$ right of $y = x,$ around the $y -axis .$
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Exercises 151
For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.
[T] $y = cos (π x), y = sin (π x), x = \frac{1}{4}, and x = \frac{5}{4}$ rotated around the This exercise requires advanced technique. You may use technology to perform the integration.
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Exercises 153
For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.
[T] $y = x^{2} - 2 x, x = 2, and x = 4$ rotated around the $x -axis .$
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Exercises 155
For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.
[T] $y = 3 x^{3} - 2, y = x, and x = 2$ rotated around the $y -axis .$
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Exercises 157
For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.
[T] $x = y^{2}, x = y^{2} - 2 y + 1, and x = 2$ rotated around the
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Exercises 159
For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.
Use the method of shells to find the volume of a cone with radius $r$ and height $h .$
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Exercises 161
For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.
Use the method of shells to find the volume of a cylinder with radius $r$ and height $h .$
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Exercises 163
For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.
Consider the region enclosed by the graphs of $y = f (x), y = 1 + f (x), x = 0, y = 0,$ and $x = a > 0.$ What is the volume of the solid generated when this region is rotated around the Assume that the function is defined over the interval
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Key terms · 1

method of cylindrical shells: a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable

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Volumes of Revolution: Cylindrical Shells: practice problems

25 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 115

For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.

[T] Bounded by the curves $y = 3 x, y = 0, and x = 3$ rotated around the $y -axis .$

Exercises 117

[T] Bounded by the curves $y = 3 x, y = 0, and x = 3$ rotated around the $x -axis .$

Exercises 119

[T] Bounded by the curves $y = 2 x^{3}, y = 0, and x = 2$ rotated around the $x -axis .$

Exercises 121