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Use a trapezoid sum with 4 sub-intervals to estimate the area between the graph of f (x) = 4-x^{2} and the x-axis on [0,2]. Is this an overestimate or an underestimate?

Since TRAP(4) is the average of LHS(4) and RHS(4), find the left- and right-hand sums and then average them.

LHS(4) = [f (0) + f (0.5) + f (1) + f (1.5)](0.5) = 6.25

RHS(4) = [f (.5) + f (1) + f (1.5) + f( 2 )](0.5) = 4.25

Then

Example 2

Let f (x) = 5x^{2} + 3x. Find TRAP(4) on [1,2].

Dividing [1,2] into 4 sub-intervals gives us sub-intervals of length 0.25 with endpoints