Description

Quadrature formulas (1)
Repeated trapezium formula:
);
with
:
Repeated Simpson s formula:

with
:
Trapezium formula for double integral
Applying succesively trapezium formula with respect to y; and with respect to x; we have:

We get

Simpson s formula for double integral
Consider the integral and the equidistant points x0;:::;x2m in [a;b]; with step respectively y0;:::;y2n in [c;d]; with step :
We apply the repeated Simpson s formula to the integral and then to the integral Algorithm:
INPUT: a,b,c,d,m,n
OUTPUT: the approximant J of the integral I
h=(b-a)/(2*n); j1=0; j2=0; j3=0 for i=0,1,…,2*n
Let x=a+i*h; hx=(d-c)/(2*m); k1=f(x,c)+f(x,d); k2=0;
k3=0; for j=1,2,…,2*m-1 y=c+j*hx;
z=f(x,y);
if j is even do k2=k2+z; else k3=k3+z;
end{if}
end{for}
l=(k1+2*k2+4*k3)*hx/3;
if (i==0)j (i==2*n) do j1=j1+l; else if i is even do j2=j2+l; else j3=j3+l; end{if}
end{if} end
J=(j1+2*j2+4*j3)*h/3
Problems
1. a) Approximate the integral
for
using trapezium formula.
b) Plot the graph of the function f and the graph of the trapezium with vertices (0;0);(0;f(0));(1;f(1)) and (1;0):
c) Approximate the integral I using Simpson s formula. 2. Approximate the following double integral

using trapezium formula for double integral, given in (1). (Exact value is:
0:4295545)
3. Evaluate the integral that arises in electrical eld theory:

for r = 110; x = 75; using the repeated trapezium formula. (Result: 6.3131) 4. Evaluate the integral

using the repeated Simpson s formula for n = 10 and 30: (Result: 0.78;0.81)
5. The volume of a solid is given by Approximate this volume applying Simpson s algorithm for double integrals for m = n = 10:
(Result: 0.178571)