MATH 293, HOMEWORK 12, due Wed Nov 20, 1996 in lecture 


*) Chapter 13 numerics. One crude way to calculate volume 
integrals numerically is as follows. Note that one need never figure out 
the limits of integration on iterated integrals to apply this method. The 
method does require nested loops (see loops examples on WWW) but the 
limits of these loops are independent of each other. 

a) put a big box around the volume.
b) divide this box into little boxes (how little depends 
    on your computer speed, your patience, the accuracy you desire, and the 
    accuracy of your computer). 
c) For each little box pick a point in the box 
   (could be the middle of the box could be one of the corners) 
d) Test to  see if the point in each box is inside the 
    volume using one or more IF statements (or something equivalent).
e) For those boxes that pass the if tests, give them credit 
    for the product of their volume and the value of the function being 
    integrated. For those points that fail one or more of the IF tests, give 
    their boxes zero credit. f) Add up the credits. This sum is the
    (crude) numerical approximation to the integral. 


1) Use the method above to generate a numerical estimate for 
    pi by calculating the volume of a sphere and comparing with the 
    known analytic formula for the volume of a sphere. 

*) You should be able to do problems 14.1.1-36. 

2) 14.1.33 (replace 'CAS' with 'MATLAB'). Use equations (1), (2), 
and (3) in section 14.1 as guidance, numerically evaluate (ie set 
up and evaluate an approximate sum). Please use this opportunity 
to build your understanding of the meaning of the definition of a 
path integral as the limit of a sum. 

*) You should be able to do most problems 14.2.1-52. You need 
a working knowledge (ie ability to set up and solve problems) of 
the words and phrases: space curve, vector field, gradient, work, 
line integral, flux, and circulation. 

3) 14.2.29.