Volumes by Slicing

April 29 2010 rlopez 1273

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Some calculus texts compute volumes of solids by the method of "slices" before they discuss the methods of disks and shells. On the other hand, there are texts that start with disks and shells, then throw in a few examples of slices. In any event, these calculations are supposed to be illustrations of how definite integration is an additive process. Unfortunately, students often get lost in the details of the individual examples, and fail to see that all these calculations are just demonstrations that definite integration is a process of addition.

In a recent Webex presentation, I was asked if Maple could do for the method of slices what its Volume of Revolution tutor does for finding the volume of a solid of revolution. As you might recall, the Volume of Revolution tutor draws a graph of the surface of revolution, writes the integral that gives the volume of the corresponding solid of revolution, and evaluates the integral. In addition, there are controls that illustrate segmentation into disks for curves rotated about a horizontal axis; and into shells, for rotation about a vertical axis. These uses of the tutor are shown in Figures 1 and 2, respectively, for the solid of revolution generated by revolving , about the -axis.

Figure 1   Solid of revolution generated by rotating , about the -axis

 

Figure 2   Illustration of the method of disks for the solid of revolution generated by rotating , about the -axis

 

 
Unfortunately, Maple does not have built-in tools for computing the volume of a solid by the method of slices. However, Maple does have sufficient primitives to assemble a solution from first principles. As the three examples included below show, the concept of adding slices is straightforward; the devil is in the details, mostly geometry, trigonometry, and algebra that is supposed to be mastered as a prerequisite to entry into the calculus course.

Problem 1

By the method of slicing, obtain the volume of a wedge cut from a cylinder of radius .

In particular, let the axis of symmetry for the cylinder lie along the -axis, the bottom face of the wedge lie in the plane , and the slanted face of the wedge lie in the plane that passes through the origin and that makes an angle  with the horizontal.

Graphics

 From the yellow triangle in Figure 1:

 

From the red triangle in Figure 1:

 

Area of red triangle:

 

Solution

Using the definite integration template from the Expression palette, enter the integral  

Context Menu: Evaluate and Display Inline

 =

 

Problem 2

By the method of slicing, obtain the volume of the solid whose base is an equilateral triangle of side , and whose cross sections are squares.  

In particular, the equilateral triangle lies in the plane , has a vertex at the origin, and an altitude along the -axis. The square cross sections are are parallel to the -plane.

Graphics

Solution

The large triangle is the base of the solid.

The vertical edge of the yellow triangle (a 30-60-90 right triangle) lies in the base of a square cross section.

Similarity of triangles then gives:

 

 

 

• 

Using the definite integration template from the Expression palette, enter the integral  

• 

Context Menu: Evaluate and Display Inline

 =

Problem 3

By the method of slicing, obtain the volume of the solid whose base in the -plane is the region bounded by the -axis, and the curves  and , and whose cross sections parallel to the -plane are equilateral triangles.

Graphics

 

Solution

Figure 2 provides a representative cross section parallel to the -plane.

The large triangle in Figure 2 is equilateral. Either half is a 30-60-90 right triangle from  is easily obtained by similarity of triangles.

 

 

 

Using the definite integration template from the Expression palette, enter the integral  

Context Menu: Evaluate and Display Inline

 =

In each of these three examples, the "calculus content" is minimal. The bulk of the intellectual energy expended is geometric and trigonometric, mathematics that a well-prepared student would have mastered in high school. In any event, the graphs and animations available in Maple, even though they are not generated by a dedicated built-in tool, are useful for all students.

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