Volume of Solids with Regular Cross-sections

Applications of Integration 2 – Solids with Regular Cross-sections

In Vino Veritas. And not only truth, but the start of an important application of the calculus. After Johannes Kepler’s (1571–1630) first wife died he decided to marry again. Naturally, he bought wine for the festivities. There was some disagreement with the wine merchant concerning how he calculated the volume of the barrel of wine. This led Kepler, a mathematician, to study the problem of finding the volume of barrels. The barrels, then as now, had circular cross sections of different diameters. His idea was to consider the wine as a stack of thin cylinders (with height of as we would say today). He then calculated the volume of each cylinder and added them together to find the volume of the barrel. There is a nice animation of the idea and more information on Kepler here.

The basic problem is this: you have a solid object whose volume you need to compute. If you can slice the object perpendicular to the axis in such a way that you get regular cross sections (i.e. similar figures) whose area you can compute, then by multiplying the area times the thickness and adding the results, you can have the volume. This kind of addition is done nicely by integration, since the volume of the slices form a Riemann sum.

You need to set up a coordinate system so that you can slice the figure into thin slices (whose width is ) and express the cross-section’s dimensions as a function of x and then the area as A(x). The result always looks like this where A(x) is the area of the cross-section:

\displaystyle \underset{n\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{A\left( {{x}_{i}} \right)\Delta {{x}_{i}}}=\int_{a}^{b}{A\left( x \right)dx}
 

Think of this as the integral of the cross-section area times the thickness

Since you are probably not allowed to have wine in your calculus class, you may want to illustrate the idea some way other than using a wine barrel. Think of a loaf of sliced bread, or a deck of cards. Find some object that the students can measure and calculate the volume, such as a tree trunk, or a statue, a Coke-Cola bottle.

Another interesting exercise is to develop (prove) the formulas for volumes that students have known and used without proof since grade school: a cone, a pyramid, a sphere, the frustum of a cone.

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