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 the integral from 0 to 4 of x squared, dx

Answer 1

f(0) = 0^2 = 0

f(1) = 1^2 = 1

f(2) =2^2 = 4

f(3) = 3^2 = 9

f(4) = 4^2 = 16

Averages the function values

0 to 1: [f(0)+f(1)]/2 = 1/2

1 to 2: [f(1)+f(2)]/2 = 5/2

2 to 3: [f(2)+f(3)]/2 = 13/2

3 to 4: [f(3)+f(4)]/2 = 25/2

Total: 44/2 = 22

Then, Technically Multiplies by rectangle width of 1

anti-derivative is (1/3)x^3

limit as x-->4: 64/3

limit as x-->0: 0

64/3 - 0 = 64/3 = 21 + 1/3