While calculus can be used to show connections between area and volume, and even be used to derive formulas,
it can also be used to apply these basic ideas. In a calculus two class, students learn about three different applications that use area and volume. The first example mentioned are integrals, and in relation with that, the Fundamental Theorem of Calculus.
In their article titled “The Fundamental Theorem of Calculus in Two Dimensions” Bennett Eisenberg and Rosemary Sullivan
explain that “Archimedes used a technique called the method of exhaustion to find the area of a region between a parabola
and a straight line.”
Then following after him, Isaac Newton developed this idea further and eventually it led to the discovery of the Fundamental
Theorem of Calculus. Both integrals and the Fundamental Theorem of Calculus are foundational parts of calculus, so it is cool to see basic concepts like area being used in the discovery of them.
Another application of area and volume in calculus is known as the “disk method”, which was mentioned above. In the disk method,
a curve is revolved about a line and then the volume of the solid created by the revolution is calculated using disks.
It is really interesting to find the volume of these revolutions as they are based off of a curve on a graph. In the desmos applet
referenced below, a student can adjust a curve on a graph and see how it affects the revolutions, and therefore the volume (pictured below).
The last application that will be mentioned is maxima and minima problems. In his book “Mathematical Foundations of Software Engineering”
by Gerard O’Regan, he states, “Maxima and minima problems refer to problems where the goal is to maximize or minimize a function on a particular interval,
where the function is continuous and differentiable on the interval, and the function does not attain its maximum or minimum at the endpoints of the interval.”
In his book O’Regan gives the example of a tank with a volume of water and using the volume formula and derivatives, one can calculate the rate at which the water
level is dropping.