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)**

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.

Not sure what kind of comments you are looking for.

I suspect a beginning calculus student will have more trouble with the physics here than with the math!

I usually start with a drag racer (kids love cars), and the relation between time, distance, velocity, and acceleration.

Suppose a car starts from rest with a uniform acceleration reaching 0 to 60 mph (88 fps) in N seconds. How far/fast does in go/accelerate? We can “derive” and “solve” the differential equation by taking ∆d/∆t to the limit stepwise, introducing the concepts of derivative and integral,

(velocity = ∆d/∆t, acceleration = ∆v/∆t) then solve for other parameter values.

Next step: birth/death process (exponential growth/decay), then harmonic oscillator, to bring in trig.