Math Forum / Mathematics / Undergraduate Math / October 2006

I am a first year calculus student and need help with the following
proejct my professor has assigned us:

-- Sticking the Tank

You have just been hired by the Environmental Protection Agency (EPA)
under the Superfund program to measure the level of toxic wastes in
buried tanks across New Mexico.  Most of these tanks are cylinders,
with their axes
horizontal.  You are to "stick the tank" by inserting a stick through
a hole in the center of the top until it touches the bottom, then
pulling it out and reading off the liquid level showing on the stick.
They have hir
ed you because you know calculus;  they have faith that you can convert
the "height on the stick" reading to "filled volume in the tank."
Assuming that the cross-sectional radius of the tank is R and its
length is L, calibrate the stick for them.  That is, convert height
showing on the stick to volume of liquid.  Check your results by doing
calculation in t
wo separate ways:

a.  Use elementary geometry and the formula for the area of a sector of
a circle 1/2R sqauared time theta (no calculus) to obtain the volume.
b.  Evaluate a definite integral that gives the filled volume in terms
of the height h on the stick.  Do not use tables!  (Suggestion: place
the origin of your coordinate system at the center of the circular
cross-section
 Make a sketch!)
c.  Show that your results for a. and b. are equal.

Since you have been so successful in such endeavors, the EPA send you
out to stick a tank which has the shape of an elliptical cylinder, i.e.
whose cross section is an ellipse instead of a circle.  The major axis
of the e
llipse is horizontal.  Calibrate the stick for them using calculus.

Extra credit:  Calibrate the stick for tilted tanks of circular and
elliptical cross sections. --

  I've found the volume geometrically easily enough for 1/2 the tank, using
the Pythagorean Theorem and the origin at the center of the circle, but I'm
not sure about the volume if the fluid is above the origin.  Also, I can't
"see" an approach for finding a definite integral.  Any thoughts on how I
can start, i.e. hints?

Jim