SOLUTION TO GAUSSIAN QUADRATURE PROBLEM #1:

Using two-term Gaussian quadrature, compute the integral from -1 to 1
of the function sin (e^t).

Using table 5.14, I get:

sin [e^0.5773] + sin [e^(-0.5773)] =  .9779 + .5324  = 1.5103

SOLUTION TO GAUSSIAN QUADRATURE PROBLEM #2:

Using two-term Gaussian quadrature, compute the integral from 0 to 2
of the function sin (e^x).

First you have to do the change of variable:

int_{0}^{2} f(x) dx = (2-0)/2 int_{-1}^{1} f[((2-0)t + 2 + 0)/2] dt

                    = int_{-1}^{1} f[t + 1] dt

                    = f[0.5773 + 1] + f[-0.5773 + 1]

		    = -.9916 + .9990

		    = .0007

Here, I'm using "int_{a}^{b} f(x) dx" to denote "the integral of f(x)
from a to b."  Sorry about the html-limited symbols...