Archive for the ‘Volume’ Category

I’ve never liked the idea of handing students formulas to use without any knowledge of where they originate. Integrals are no different; I believe its easier for students to grasp the idea of integration by setting up Riemann sums first. From this perspective, students can focus on just one “member” in the group of infinitely numerous elements and, from that perspective, build representations for area, volume, force, work, etc. Once students have an expression for that single member, the more abstract integral can be more easily conceptualized.

The entry below describes multiple components working in unison to determine the work required to lift water up and over the edge of a container. The applet provided allows students to interact with two variables (depth & lower radius) and observe how changes to those influence the corresponding algebraic representations. This scenario of lifting columns of water is directly related to the work required to raise a length of chain upwards by a given distance; an example of this follows as well.


Cylindrical Container



Conical Container



Click on the link provided here to explore a variable column of water.


I’ve added the following scenario for additional contrast/comparison. This example illustrates rectangular slabs of water being lifted over the upper edge of a v-shaped trough.

V-Shaped Trough


Click on the link provided here to better visualize work required to remove rectangular slabs from a container.


Lifting a Chain



Click on the link provided here to lift a chain.


Work as a Function of Distance (y)

Work over Distance


Click on the link provided here to explore dW/dy.


I wanted work described in terms of “ft•lb” here since students have been exposed to that terminology with respect to torque (rotational analog of work). A later entry will be focused on this which will, in turn, open the door to an exploration into horsepower.


Thanks for reading.

As in the case of the cone, volume of the pyramid can be determined in various ways. The version here once again dabbles with a multi-variable scenario to give students some variation on a familiar theme.

Pyramid Volume2

Click on the link to interact with the notion of infinitesimals as applied to pyramid volume.

Refer to volume of cone for helpful comparison.


Thanks for reading.

This entry will no doubt get further under the skin of those who contend that I’m pushing things too far (and that is exactly why I’m publishing it).  This is a natural extension of the 2-dimensional analog in which circle area was derived. Cylindrical coordinates are identical to polar coordinates with a 3rd dimension thrown in. Directly below, the vertical dimension is governed by a linear function which results in a cone.

In the spirit of consistency, I’ve included a second example with which to draw comparisons to the first. Everything is identical between the two cases with one exception; the function which governs the vertical dimension in example two is that of a semi-circle.


Volume of a Cone

VolumeCone Cylindrical

Interact with cylindrical coordinates to see how the 3-dimensional sector changes.


Volume of a Sphere

VolumeSphere Cylindrical


Thanks for reading.

I’m snooping around multi-variable calculus with this entry. Presentation of this in class will depend on circumstances (time and where students are). I will nevertheless post a link to this on Google classroom and let students explore if they so desire.

Cone volume (multivariable)

Explore the volume of a cone from a different perspective.

Check out volume of pyramid as well.

Thanks for looking.