MCS471s96 Practice Problems for Numerical Integration with Some Answers
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CAUTION: These problems are just for practice and are not to be handed in.
         Some of these old exam problems and answers may not be relevant to
         your current course.   Caveat Usor!

      In the following problems, use "EXAM PRECISION: chop to 4 
significant digits only when you write an intermediate or final answer down
and continue calculations with those numbers recorded.
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Background Reading: Gerald and Wheatley, 
Chapter 4 Topics

4.2 Derivatives from Difference Tables 
4.3 Higher-Order Dervatives
4.4 Extrapolation Techniques
4.5 Newton-Cotes Integration Rules 
4.6 Trapezoidal Rule 
4.7 Simpson's Rule 
4.9 Gaussian Quadrature 
4.10 Adaptive Integration 

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     0. For problems in the Gerald and Wheatley text, 
try Chapter 4 Exercises (pp. 384-392, 5th ed.)
     #1,6,7,11,14,31,32,33,36,37,44,45,51,58,61.
 
     1.   In parts (a) to (d), approximate  
                                    1
                                   S  f(x) dx,
                                    0
where   
             {  (1+x)  on  (0,0.5)
      f(x) = {                        ,
             {  (2-x)   on  (0.5,1.0)

according to the following simple rules:
          (a) Trapezoidal Rule on (0,1.).                    (Ans.:  1.000)
          (b) Simpson's Rule on (0,1.).                      (Ans.:  1.333)
          (c) Midpoint Rule on (0,1.).                       (Ans.:  1.500)
          (d) Double Trapezoidal Rule on (0,0.5) + (0.5,1).  (Ans.:  1.250)
 
     2.   In parts (a) to (e), approximate
                                    +1
                                   S  ln(1+x**2) dx,
                                    -1
for the following rules:
          (a) Simpson's Rule with 2 subintvals (SR3).        (Ans.:  0.4620)
          (b) Midpoint Rule with 4 subintvals (MR4).         (Ans.:  0.5069
                                                                  or 0.5068)
          (c) Trapezoidal Rule with 4 subintvals (TR5).      (Ans.:  0.5697)
          (d) Simpson's Rule with 8 subintvals (SR9).        (Ans.:  0.5278)
          (e) Two-point Gaussian Rule (GR2).                 (Ans.:  0.5753)
          (f) About how many subintervals  (n?)  are needed for an n-panel
composite Trapezoidal Rule so that the absolute value of the global 
approximation error is less than  1.e-8,  given that   abs(f"(x)) <= 3  on the
integration interval  (-1.,+1.).  Round the final answer to the nearest 
integer.  {<= means less than or equal}                      (Ans.:  14142)
 
     3.   Approximate the integral:  
                                3 
                               S  exp(x)*sqrt(x) dx 
                                0 
using 7 points and Simpson's composite rule.  Minimize the number of function 
evaluations, multiplications and additions.       
                                      (Ans.:  27.45 or 27.43 with extra chops)

     4.   Estimate how many function evaluations  (i.e., points used) are
needed for an efficient, composite Simpson's Rule, so that the absolute 
value of the global approximation error is  <= 0.5e-3.  The interval of 
integration is (-3,2),  -2**P/P!  <= (d/dx)**p*f(x)  <= 2**(-p)/p!  for all
x and p;  and the chopping error is assumed to be negligble.  {<= means less
than or equal}                        (Final Ans.:  15 function evaluations.)
 
     5.   Approximate the integral:
                                     4 
                               I =  S  (sqrt(x)/exp(x))  dx  
                                     1 
using a 7 point Simpson's rule.  Minimize the number of function evaluations
and multiplications in calculating the composite rule.  
(Ans.:  
      x =      (1.000  1.500  2.000  2.500  3.000     3.500     4.000)
   f(x) =(4ch) (0.3678 0.2732 0.1913 0.1297 0.8623e-1 0.5645e-1 0.3663e-1),
I =(4ch) (0.3678+4*(0.2732+0.1297+0.5649e-1)+2*(0.1913+0.8623e-1)+0.3663e-1)/6.
  =(4ch) (0.4661,0.4660,0.4663), depending on the chopping sequence.)

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