Approximations with 20 Subdivisions

By pointing and clicking on Evaluate Numerically we obtain

L_f(1,5,20)=65.52

R_f(1,5,20)=81.52

T_f(1,5,20)=73.52

M_f(1,5,20)=73.24

S_f(1,5,20)=73.33333.

And we can see at once that the midpoint sum is better than the trapezoidal sum which, in turn, is much better than the left and right sums. We see also that the Simpson sum is the best of all. As a matter of fact, the exact value of the Simpson sum is which is exactly equal to the integral. It can be proved that the Simpson sum is always exactly correct when the function being integrated is a polynomial of degree 3 or less.