Trapezoidal Rule

Question

Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for Int [0,4] x²dx.

Answer 1

The lower sum for this partition is (0+1+4+9) = 14 and the upper sum is (1+4+9+16) = 30

Average of lower and upper sum = 22

The trapezoidal rule is: .5*(1 + (1+4) + (4+9)  + (9+16)) = 44/2 = 22

You can also see this if you write the three expressions one under the other.