Trapezoidal rule calculus question

Question

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

Answer 1

Δx = (b-a)/4 = 4/4 = 1

upper Area = (1)∑f(x_i+i) = (1)∑(x_i+i)²

lower Area = (1)∑f(x_i) = (1)∑(x_i)²

Average = (upper+lower)/2= (∑(x_i+i/4)² +∑(x_i)²)/2 = (1/2)∑((x_i+i)²+(x_i)²)

Trapezoidal = (1)∑(f(x_i+i) +f(x_i))/2 = (1/2)∑(x_i+i)²+(x_i)²) = Average

Upper = 1²+2²+3³+4² = 30

Lower = 0²+1²+2²+3² = 14

Average = (30+14)/2 = 44/2 = 22

Trapezoidal = (1/2)(1 + 0 + 4 + 1 + 9 + 4 + 16 + 9) = 44/2 = 22