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  Modeling the real world: The concept of volume is utilized in a wide variety of applications including the physical sciences and all of the engineering disciplines. For example, in constructing a highway one uses survey data to approximate the amount of earth to be moved. The definite integral can be used to derive formulas to make such calculations.

Example 1. Volume of solid with circular base and square cross-sections.

The following two clips are taken from an on-line exam given at the University of Pittsburgh.

In this first clip the author presents the application of cross-sections to a solid with a circular base and square cross-sections.

Below is a photograph of two models of this solid. The one on the left has alternating color woods to show the slices, while the one on the right is cut with pieces separated to show a sample slice. The models were made by Foster Manufacturing Company.

The steps involved include:

1. View the solid with a circular base of radius r.
2. Regard each of the cross-sections as a square perpendicular to the base.
3. Compute the volume of the solid by integrating from -r to r the area of the cross-section.
4. Hint: Draw a big circle of radius r and for -r < x < r draw the vertical segment given by the intersection of the circle with the perpendicular line that goes through x. Call the length of this segment .
5. The area of the cross-section is .
6. Note that is on the circle .
7. Thus, .
8. We then apply the definite integral to find the volume by integrating the area of the cross-section from -r to r.

In the next screen we have the final evaluation for the volume.

In this example we consider an application of the definite integral to the computation of work. (*From Discovering Calculus by Levine and Rosenstein, Jr., McGraw-Hill).

Solution:

Note: In this problem the force is constant but different "parts" of the water are moved different distances. Note that the water at the bottom must travel farther than the water at the top. The solution will require a Riemann sum process similar to the volume example above.

1. First we introduce a coordinate system with the y-axis down the center of the tank and the origin at the center of the bottom.
2. Next we slice the water into thin disks of thickness, D y. (This step mirrors the cross-section process.)
3. The volume of water in the slice = p r² D y.
4. The mass of the water in this slice = r p r^{2 D} y; r is the density of water (approximately 1 gram/cc).
5. Let y_j be the y-coordinate of any point in the j ^th slice.
6. Thus, the water in the j ^th slice must be moved a distance ( h - y_j ) .
7. The work required to pump the water in the j ^th slice out of the tank is

W_j = mass x acceleration x distance moved = r g p r² ( h - y_j ) D y, (g = acceleration due to gravity).

8. Thus, the total work to pump out all the water is approximately



 

9. Note: The above sum is a Riemann sum for the function f(y) = r g p r² ( h - y )
10. We now take the limit as and noting that r , g, p and r² are constants we obtain

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USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil