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Let . Use a trapezoid sum with 3 sub-intervals to estimate the area between f and the x-axis on [0,9]. Is this an overestimate or underestimate?
We divide [0,9] into 3 sub-intervals of width 3.The first trapezoid has
The second trapezoid has
The third trapezoid has
Adding up the areas of the trapezoids, we get
71.625 + 58.125 + 31.135 = 160.875.
This is an underestimate, since the trapezoids don't cover the whole area between f and the x-axis on [0,9].
The table below shows partial values of the function f (x). Use a trapezoid sum with 4 sub-intervals to estimate the area between the graph of f and the x-axis on [0,1].
Each sub-interval has width 0.25. First trapezoid, on [0,0.25]:
Second trapezoid, on [.25,.5]:
Third trapezoid, on [.5,.75]:
Fourth trapezoid, on [0.75,1]:
We add the areas of the trapezoids and get
0.75 + 1.25 + 1.875 + 2.5 = 6.375.