here's an example showing how to use maple to evaluate large left 
and right hand sums.  this is problem 10 from section 5.2 on page 235.

> f := x-> sin(x^2);

                                         2
                          f := x -> sin(x )


first, let's do the n=2 case.  

> n := 2;

                                n := 2


it's important to type ":=", not "=".  The former command tells Maple
that n should be 2 for the rest of the session, not for just one line.

next, let's set our delta_t.  the size of delta_t is the length of the interval
(in this case, 3-2), divided by n:

> delta_t := (3-2)/n;

                            delta_t := 1/2


now, we need to evalate the left hand sum.  

> sum( f(2+ (i*delta_t) )*delta_t, i=0..n-1); 

                      1/2 sin(4) + 1/2 sin(25/4)



I had hoped for a number, not something symbolic.  to force maple to give me a nice
number, i need to type evalf(%).  The command evalf tells maple to EVALuate the
expression as a Floating point number.  The "%" is a shorthand for the previous output,
i.e., the previous line of blue output text.

> evalf(%);

                             -.3949908560


so far, so good.  i next need to do the right hand sum.  to do this, all i need
to do is change the limits on the left hand sum

> sum( f(2+ (i*delta_t) )*delta_t, i=1..n):
> evalf(%);

                             .1894696343

> 

Thus, for n=2, the left and right hand sums are and -.395 and .189.   

To do this for different n's, you can go back, change the value of n, and then 
repeatedly hit enter and each next line of Maple will evaluate with the
new value of n.  For large n's you'll get an impressively long list of terms.
If you don't want these filling up your screen you can wrap an evalf
around the sum term.  You could also end the line with a colon instead
of a semi-colon; ending with a colon supresses blue output.