# Assignment C

Your assignment is to find the volume of a wine fermentation tank made by Sonoma Stone. When converted to the metric system (because wine bottles have a volume of 750 milliliters), the boundary of the central cross section of the interior of the tank is very nearly $$0.0002017404542x^2+ \label{eq:egg} \frac{0.0001303290910y^2}{0.9520444748+\alpha y} = 1.$$ All distances are measured in centimeters. In other words, the interior of the tank is interior of the surface described by rotating that curve about the $y$ axis. The volume of that region can be found as an integral. I.e. we are evaluating an integral to find the volume of the interior of a surface of rotation. For the real tank, $\alpha \approx 0.005$.

1. Create a Python/Sympy script to evaluate the volume for any value of $\alpha$ between 0 and 0.02. Comment the file liberally.

2. Figure 1
3. Find a way in your script to plot the cross section of the tank when $\alpha =0.005$. A sample appears in Figure 1.
4. Plot the value of the volume of the tank for values of $\alpha$ between 0 and 0.02.
5. Write a LaTeX document describing how to find the volume of the tank. Remember that this is a mathematical discussion - you will use mathematical notation to describe the process - not computer or Python notation. Moreover, the instructor has no interest in a description of the computational steps in finding the volume - he wants to know how you solved mathematically for the various components of the integral. Obviously, your document will include some astute observations about the way that the volume depends on $\alpha$, and so on.
6. Make sure to tell how many bottles of wine one expects to get out of the tank when $\alpha =0.005$. Do not try to include the solution for all values of $\alpha$, unless it were in an appendix.
7. Make sure to include the pictures of the cross-section of the tank that you created, as well as the one for variation of volume with $\alpha$.
8. You are likely at some point to want to solve for $$x$$ or $$y$$. Recall that you can do that in Sympy using e.g. solve(expression,x). That solves the equation expression=0 for the variable x.

This assignment is worth 60 points and is due at 9 AM on Monday, 10 December. It is turned in when a message containing four attachments appears in the instructor's email inbox. The attachments are a) the Python .py file; b) the .tex file; and c) the image for your cross section; and d) the image of volume as a function of alpha. These files are all small - do not archive or compress them.

Hint: Sometimes people experience the following terrifying error.

 sympy.polys.polyerrors.CoercionFailed: can't convert DMP([1, 0], ZZ, ZZ[_b1]) of type ZZ[_b1] to RR

This seems to indicate that Sympy has tried to evaluate the integral in the complex plane, and failed. There are a couple of ways to fix this, but perhaps the simplest is to make a little function that evaluates the integrand:
def integrand(t):
return e.g. x value in terms of y

Then use the numerical integration function from Sympy to evaluate the integral.
volume = sp.mpmath.quad(integrand,(lower_limit,upper_limit))