Let . Use a trapezoid sum with 4 sub-intervals to estimate the area between the graph of f and the x-axis on [1, 3].
Answer
We divide [1,3] into sub-intervals of length .5. We need to know the following values of f:
Then
Example 2
Let g(x) = 3x + 2. Use a trapezoid sum with 3 sub-intervals to estimate the area between the graph of g and the x-axis on [0, 6].
Answer
We divide [0, 6] into 3 sub-intervals of size 2, with endpoints x = 0, 2, 4, and 6.
Example 3
Values of the function f are given in the table below. Use a trapezoid sum with 8 sub-intervals to estimate the area between f and the x-axis on [2, 10].
Answer
We don't need all the information that's given in the table. Since we're asked about the interval [2, 10], we don't care what f(0) and f(1) are.
Dividing [2, 10] into 8 sub-intervals gets us sub-intervals of length 1, so we do need all the values given in the table except f(0) and f(1).
Example 4
Values of the function f are shown in the table below. Use a trapezoid sum with the sub-intervals suggested by the table to estimate the area between f and the
x-axis on [0, 10].
Hint
Average the left-hand sum and right-hand sum.
Answer
The sub-intervals are [0,1], [1, 3], [3, 6], [6, 7], [7, 8], and [8, 10]. Since they have unequal lengths, it's probably easiest to find the left-hand sum and right-hand sum, then take their average.
. Use a trapezoid sum with 4 sub-intervals to estimate the area between the graph of f and the x-axis on [1, 3].
