Problem in understandind sultion in numerical analisys

[nhrock3]

A)
calculate $$I=\int_{0}^{1}\cos(\frac{\pi}{2}x^2)$$
by trapeze methos in those pannels
m=1,2,4,8
?

B)
what is the best aproximation we can get from part A of the question?

solution:
the solution for part A is
http://i47.tinypic.com/2vmh4b9.jpg

my problem start with the solution of part B

http://i45.tinypic.com/b4jorq.jpg

how they find $$T_{2,1}$$ for example??
how each index plays a role in the formula
by what formula they work on??

how they find out that the best approximation is$$T_{3,1}$$
??

 

[lippyka]

The best approximation we can get from part A is the value of T3,1, which is 0.9727. This value was obtained by using the trapezoidal method, where the integral was divided into n subintervals with an equal width of h = 1/m, where m is the number of subintervals. To find T3,1, we need to calculate the area of the trapezoids formed by the function values at the endpoints of each subinterval and add them together. The formula for calculating T3,1 is: T3,1 = (h/2)*(f(x0) + 2*f(x1) + 2*f(x2) + 2*f(x3) + f(x4)), where h = 1/4 and x0, x1, x2, x3, and x4 are the endpoints of the subintervals.