Volume of a Parabolic Vase
Introduction
The following procedure is how to find the volume of a parabolic vase. This is a vase where the "sides" (for the lack of a better term) form a parabolic curve. Each side of a vase has a smooth bump, away from the center, with one peak. Imagine if that you can turn a vase on its side. The result is like the illustration above.
Note: The following procedure works with vases where the "sides" curved towards the center, rather than away.
Steps
Using calculus, we can approximate the the volume by taking the following steps:
1. Measure the length of the vase. Let x be this length.
2. Find the "center line" of the vase. From the center line, find the distance from the center line to the end of the case. Do this for both ends of the vase, and at half of the length. Get the best measurement you can. From one end to the other, these are lengths y_0, y_1, and y_2, respectively.
3. Use the data to form the parabolic curve. Find a, b, and c.
4. With a, b, and c determined, plug in a, b, c, and x and calculate the following integral.
Most graphing calculators can handle this task easily.
Note: If there is an official term for these type of vases, I would like to know. Many thanks in advance.
Example
Find the volume of a parabolic vase 4 inches long with the following: (see the illustration "Parabolic Vases with measurements").
So we have these points to form the parabolic equation with:
(0,4), (2,6), (4,4)
Then:
which leads to:
c = 4
b = 2
a = -0.5
The integral is:
Calculating the integral gives an approximate area of 134.041287 in^2.
Found at the Swap Meet: Casio fx-78
Several Sundays ago I purchased this calculator at the Azusa Swap Meet. This calculator, made in 1982, is not only basic, but quite small. It still works.
Have a great day, Eddie
This blog is property of Edward Shore. © 2012