M2 Definite Integration

Divide into n subintervals:
0 = x0 < x1 < x2 < ... < xn = 3
x(i + 1) - xi = △x = (3 - 0)/n = 3/n for i = 0, 1, 2, ..., n - 1
-> xi = 3i/n

height of subintervals
= f(xi)
= f(3i/n)
= 3i/n + 3

area of subintervals
= f(xi)△x
= 9i/(n^2) + 9/n

Required answer
= Sum of areas of subintervals
= (i = 0, n - 1)∑ [9i/(n^2) + 9/n]
= [9/(n^2)][1 + 2 + ... + (n - 1)] + n(9/n)
= [9/(n^2)]{[1 + (n - 1)]n}/2 + 9
= [9/(n^2)](n^2)/2 + 9
= 9/2 + 9
= 27/2