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?
Answer
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].
Example 2
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].
. 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?
