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.