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 + 31.125 = 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 [0.25, 0.5]:
Third trapezoid, on [0.5, 0.75]:
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.
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