Volume by Slicing Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Integrals can calculate volume as well as area. One approach is to think of a 3-dimensional region as being made up of many thin slices, and then to add up the volumes of the slices in a Riemann sum. Section 6.2 of our textbook explores several forms of this idea. The following examples and questions will help make the basic idea more concrete.

Triangular Pyramid

Imagine a pyramid with a right-triangle base, lying on its side on the x axis so that it’s point is at the origin and its base is at x = 2. The two legs of the right triangle that form the base are each 1 unit long, and the pyramid tapers linearly to its point:

See if you can use the slicing method to calculate the volume of this shape.

Parabolic “Pyramid”

Similar to the previous shape, except that instead of the sides tapering linearly to the point, they follow parabolic curves. In particular, at each x value, the sides of the triangular cross-section have length 2√x:

What is the volume of this shape?

Frustum

A frustum is a pyramid with its top cut off, i.e., a tapered shape with a rectangular cross section that doesn’t come to a point. For example, here is frustum lying on its side:

The frustum extends from x = 1 to x = 3; at any given x coordinate between those bounds, the width of the frustum (awkwardly labeled “L” in the figure) is x, and the height is x/2. What is the frustum’s volume?

Curved “Frustum”

A shape like the frustum above, except that the height at any x value is given by H = e^x2, and the frustum extends from x = 0 to x = 1. The width at each x value is still x:

What is this shape’s volume?