Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 4 of x squared, dx

Question

Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for 



Answer 1

n = 4, Δx=(4-0)/4 = 1

LHR = 1[0+1+4+9] = 14

RHR = 1[1+4+9+16] = 30

(LHR+RHR]/2 = (14+30)/2 = 22

Trap = (Δx/2)[f(0) + 2f(1)+2f(2)+2f(3)+f(4)] = (1/2)[0 + 2(1)+2(4)+2(9)+16] = (2+8+18+16)/2 = 1+4+9+8 = 22