Midpoint Sums with Shortcuts Examples

Let f(x) = 3x. Use a midpoint sum with 4 sub-intervals to estimate the area between f and the x-axis on [0, 2].

Dividing [0, 2] into 4 sub-intervals gives us sub-intervals of length 0.5 and midpoints

0.25, 0.75, 1.25, 1.75

Since the distributive property says that

f (0.25)(0.5) + f (0.75)(0.5) + f (1.25)(0.5) + f (1.75)(0.5)

and

[f (.25) + f (0.75) + f (1.25) + f (1.75)](0.5)

are the same thing, we can evaluate the one that only requires multiplying by 0.5 once: