Volume Examples

Volume and Area Examples

Here is a Maple Worksheet that investigates the volume of a solid of revolution including looking at sums of volumes of cross section cylinders. In the worksheet the interval is [ -1 , 2 ] and each x_i is the midpoint of the ith subinterval. Thus if there are n equal length subintervals then the length of each subinterval will be 3/n and each x_i will be given by -1 + (3/n)(i - 0.5). Quicktime Animation

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this web site.

QuickTime free download.

Example 1

Find the volume of the solid formed by revolving the region bounded by the graphs of the equations given below about the x-axis. DPGraph Picture

Example 2

Find the volume of the solid formed by revolving the region in Example 1 about the line y = -2.

Click here to see a picture of the two surfaces that form the sides of the solid. DPGraph Version

Example 3

Find the volume of the solid formed by revolving the region bounded by the graphs of the equations given below about the y-axis. DPGraph Picture

Example 4

Find the volume of the solid formed by revolving the region bounded by the graphs of the equations given below about the x-axis. DPGraph Picture

Example 5

Find the volume of the solid whose base is bounded by the graphs of the equations given below and whose cross sections are equilateral triangles perpendicular to the x-axis.

Example 6

Shell Method Example: Find the volume of the solid formed by revolving the region bounded by the graphs of the equations given below about the line x = -1.

Example 7

Find the volume of the solid formed by revolving the region bounded by the graphs of the equations given below about the x-axis. DPGraph Picture

Example 8

Shell Method Example: Find the volume of the solid formed by revolving the region bounded by the graphs of the equations given below about the line y = -2.

Example 9: Volume of a Solid--Disk Method

The picture at the right shows the graph of the curve being revolved about the x-axis.

DPGraph Picture 1--Use the scrollbar and activate "b" to revolve the region about the x-axis and form the solid.

DPGraph Picture 2--Use the scrollbar and activate "b" to revolve the region about the x-axis and form the solid. This time the solid will be transparent. Use the scrollbar and activate "a" to look at cross section disks.

DPGraph Picture 3--Use the scrollbar and the x slice feature to look at cross sections.

DPGraph Picture 4--Use the scrollbar and activate "a" to look at cross sections between two parallel planes that are perpendicular to the x-axis.

DPGraph Picture 5--Use the scrollbar and activate "a" to look at lateral surfaces of thick cross section disks.

DPGraph Picture 6--Use the scrollbar and activate "a" to look at cross section disks only (no solid). You can activate "b" to draw in the solid.

DPGraph Picture 7--The same as DPGraph Picture 6 except that "b" changes with time to produce an animated drawing of the solid.

Example 10: Volume of a Solid--Method of Cylindrical Shells

DPGraph Picture of the solid of revolution--Use the scrollbar to vary the value of "a" and see various "cylindrical shells". Each "shell" has a "thickness" of 1/2 unit.

DPGraph Picture2--Use the scrollbar to vary the value of "a" again to see the "shells". In this case the "shells" are thin.

Example 11: Volume of a Solid Using a Cross Section Area Function

Find the volume of the solid whose base lies in the xy-coordinate plane and is bounded by the graphs of y = x² and y = 4 and vertical cross sections perpendicular to the y-axis are squares.

DPGraph Picture 1--Use the scrollbar to activate "a" to look at individual cross section squares.

DPGraph Picture 2--Use the scrollbar to activate "a" to look at individual cross section squares inside a transparent image of the solid.

DPGraph Picture 3--This is an automated animation of DPGraph Picture 2.

DPGraph Picture 4--This is similar to DPGraph Picture 2 except that one side of the solid is left "open" and the solid is not transparent. Use the scrollbar to activate "a" again.

Two Area Examples

Example 12: Area Between Two Curves

Find the area of the region bounded by the graphs of the functions f(x) and g(x) given below.

Maple Worksheet Similar Example--Audio/Video--Approximating on a TI-84

Example 13: Area Between Two Curves-Approximation

Maple Worksheet

Follow this link to additional example's at Scott Rickman's (Seminole State instructor) web site. These examples include volume using a cross section area function, by the disk method, washer method, and shell method, and surface area along with DPGraph illustrations. He has some extremely clever graphics that are fun and instructive and gave me the ideas for some of my graphics.

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu