- Dividing the interval [0, 12] into 3 equal sub-intervals gives us sub-intervals of width 4. We find the function values we need by looking them up in the table.
Sub-interval [0, 4]: The midpoint of this sub-interval is 2. This means the height of the rectangle is *g*(2) = 5. Since the width is 4, the area of this rectangle is 5(4) = 20. Sub-interval [4, 8]: The midpoint of this sub-interval is 6. This means the height of the rectangle is *g*(6) = 11. Since the width is 4, the area of this rectangle is 11(4) = 44. Sub-interval [8, 12]: The midpoint of this sub-interval is 10. This means the height of the rectangle is *g*(10) = 25. Since the width is 4, the area of this rectangle is 25(4) = 100. Adding the areas of these rectangles, we estimate the area between *g* and the *x*-axis on [0, 12] to be 20 + 44 + 100 = 164. - If we divide the sub-interval [0, 12] into 4 equal sub-intervals we get the sub-intervals
[0, 3], [3, 6], [6, 9], and [9, 12]. The midpoints of these sub-intervals are, respectively, 1.5, 4.5, 7.5, and 10.5. The table doesn't tell us what *g* is at those places. We can't use the table to find a midpoint sum with 4 equal sub-intervals. - If we divide the sub-interval [0, 12] into 6 equal sub-intervals we get the sub-intervals
[0, 2], [2, 4], [4, 6], [6, 8], [8, 10], [10, 12]. The midpoints of these sub-intervals are, respectively, 1, 3, 5, 7, 9, 11. Since the table does tell us what *g* is at those midpoints, yes, we could use this table to take a midpoint sum with 6 equal sub-intervals. | |