the intersection of a plane and a solid object

a special case of the slicing method used with solids of revolution when the slices are disks

a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume

a solid generated by revolving a region in a plane around a line in that plane

a special case of the slicing method used with solids of revolution when the slices are washers

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Calculus Volume 1 · 6.2 Determining Volumes by Slicing

Determining Volumes by Slicing: practice problems

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

Exercises 63
For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.
A pyramid with height 6 units and square base of side 2 units, as pictured here.
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Exercises 65
For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.
A tetrahedron with a base side of 4 units, as seen here.
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Exercises 67
For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.
A cone of radius $r$ and height $h$ has a smaller cone of radius $r / 2$ and height $h / 2$ removed from the top, as seen here. The resulting solid is called a frustum.
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Exercises 69
For the following exercises, draw an outline of the solid and find the volume using the slicing method.
The base is a triangle with vertices $(0, 0), (1, 0),$ and $(0, 1) .$ Slices perpendicular to the x-axis are semicircles.
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Exercises 71
For the following exercises, draw an outline of the solid and find the volume using the slicing method.
The base is the region under the parabola $y = 1 - x^{2}$ and above the $x -axis .$ Slices perpendicular to the $y -axis$ are squares.
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Exercises 73
For the following exercises, draw an outline of the solid and find the volume using the slicing method.
The base is the area between $y = x$ and $y = x^{2} .$ Slices perpendicular to the x-axis are semicircles.
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Exercises 75
For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.
$y = 2 x^{2}, x = 0, x = 4, and y = 0$
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Exercises 77
For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.
$y = x^{4}, x = 0, and y = 1 for x \geq 0$
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Exercises 79
For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.
$y = sin x, y = cos x, and x = 0$
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Exercises 81
For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.
$x^{2} - y^{2} = 9 and x + y = 9, y = 0 and x = 0$
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Exercises 83
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
$y = 2 x^{3}, x = 0, x = 1, and y = 0$
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Exercises 85
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
$y = 4 - x^{2}, y = 0, and x = 0$
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Exercises 87
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
$x = sec (y) and y = \frac{π}{4}, y = 0 and x = 0$
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Exercises 89
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
$y = 4 - x, y = x, and x = 0$
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Exercises 91
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.
$y = x^{2} and y = x + 2$
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Exercises 93
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.
$y = 4 - x^{2} and y = 2 - x$
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Exercises 95
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.
$y = x and y = x^{2}$
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Exercises 97
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.
$y = 1 + x^{2} and y = 4 - x^{2}$
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Exercises 99
For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
$y = x + 2, y = 2 x - 1, and x = 0$
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Exercises 101
For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
$x = e^{2 y}, x = y^{2}, y = 0, and y = ln (2)$
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Exercises 103
For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
Yogurt containers can be shaped like frustums. Rotate the line $y = \frac{1}{m} x$ around the y-axis to find the volume between $y = a and y = b .$
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Exercises 105
For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
Rotate the ellipse $(x^{2} / a^{2}) + (y^{2} / b^{2}) = 1$ around the y-axis to approximate the volume of a football.
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Exercises 107
For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.
What is the volume of the Bundt cake that comes from rotating $y = sin x$ around the y-axis from $x = 0$ to $x = π ?$
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Exercises 109
For the following exercises, find the volume of the solid described.
The base is the region enclosed by the generic ellipse $(x^{2} / a^{2}) + (y^{2} / b^{2}) = 1.$ Slices perpendicular to the x-axis are semicircles.
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Exercises 111
For the following exercises, find the volume of the solid described.
Find the volume common to two spheres of radius $r$ with centers that are $2 h$ apart, as shown here.
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Exercises 113
For the following exercises, find the volume of the solid described.
Find the volume of a sphere of radius $R$ with a cap of height $h$ removed from the top, as seen here.
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Review Exercises 437
True or False? Justify your answer with a proof or a counterexample.
The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.
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Review Exercises 439
For the following exercises, use the requested method to determine the volume of the solid.
The volume that has a base of the ellipse $x^{2} / 4 + y^{2} / 9 = 1$ and cross-sections of an equilateral triangle perpendicular to the $y -axis .$ Use the method of slicing.
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Review Exercises 441
For the following exercises, use the requested method to determine the volume of the solid.
$x = y^{2}$ and $x = 3 y$ rotated around the y-axis using the washer method
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Review Exercises 443
For the following exercises, find
the area of the region,
the volume of the solid when rotated around the x-axis, and
the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.
$y = x^{3}, x = 0, y = 0, and x = 2$
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Review Exercises 445
For the following exercises, find
the area of the region,
the volume of the solid when rotated around the x-axis, and
the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.
[T] $y = ln (x) + 2 and y = x$
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Review Exercises 447
For the following exercises, find
the area of the region,
the volume of the solid when rotated around the x-axis, and
the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.
$y = 5 + x,$ $y = x^{2},$ $x = 0,$ and $x = 1$
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Review Exercises 449
For the following exercises, find
the area of the region,
the volume of the solid when rotated around the x-axis, and
the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.
Find the mass of $ρ = \frac{1}{x ^{2} + 1}$ on a disk centered at the origin with radius $4.$
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Review Exercises 459
For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings.
Find the volume of the catenoid $y = cosh (x)$ from $x = - 1 to x = 1$ that is created by rotating this curve around the $x -axis,$ as shown here.
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Key terms · 5

cross-section: the intersection of a plane and a solid object
disk method: a special case of the slicing method used with solids of revolution when the slices are disks
slicing method: a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume
solid of revolution: a solid generated by revolving a region in a plane around a line in that plane
washer method: a special case of the slicing method used with solids of revolution when the slices are washers

Learn 6.2 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 6.2 in the textbook · This section on openstax.org

Determining Volumes by Slicing: practice problems

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

Exercises 63

For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A pyramid with height 6 units and square base of side 2 units, as pictured here.

This figure is a pyramid with base width of 2 and height of 6 units.

Exercises 65

For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A tetrahedron with a base side of 4 units, as seen here.

This figure is an equilateral triangle with side length of 4 units.

Exercises 67

For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A cone of radius $r$ and height $h$ has a smaller cone of radius $r / 2$ and height $h / 2$ removed from the top, as seen here. The resulting solid is called a frustum.

This figure is a 3-dimensional graph of an upside down cone. The cone is inside of a rectangular prism that represents the xyz coordinate system. the radius of the bottom of the cone is “r” and the radius of the top of the cone is labeled “r/2”.

Exercises 69

For the following exercises, draw an outline of the solid and find the volume using the slicing method.

The base is a triangle with vertices $(0, 0), (1, 0),$ and $(0, 1) .$ Slices perpendicular to the x-axis are semicircles.

Exercises 71

For the following exercises, draw an outline of the solid and find the volume using the slicing method.

The base is the region under the parabola $y = 1 - x^{2}$ and above the $x -axis .$ Slices perpendicular to the $y -axis$ are squares.

Exercises 73

For the following exercises, draw an outline of the solid and find the volume using the slicing method.

The base is the area between $y = x$ and $y = x^{2} .$ Slices perpendicular to the x-axis are semicircles.

Exercises 75

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$y = 2 x^{2}, x = 0, x = 4, and y = 0$

Exercises 77

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$y = x^{4}, x = 0, and y = 1 for x \geq 0$

Exercises 79

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$y = sin x, y = cos x, and x = 0$

Exercises 81

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$x^{2} - y^{2} = 9 and x + y = 9, y = 0 and x = 0$

Exercises 83

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$y = 2 x^{3}, x = 0, x = 1, and y = 0$

Exercises 85

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$y = 4 - x^{2}, y = 0, and x = 0$

Exercises 87

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$x = sec (y) and y = \frac{π}{4}, y = 0 and x = 0$

Exercises 89

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$y = 4 - x, y = x, and x = 0$

Exercises 91

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y = x^{2} and y = x + 2$

Exercises 93

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y = 4 - x^{2} and y = 2 - x$

Exercises 95

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y = x and y = x^{2}$

Exercises 97

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y = 1 + x^{2} and y = 4 - x^{2}$

Exercises 99