We're driving along from right coast to the left coast, and now it's time to take a rest stop at the midpoint sum. Grab some snacks before continuing on. We recommend all flavors of Sun Chips and wasabi almonds.
A midpoint sum is similar to a left-hand sum or right-hand sum. For a midpoint sum, the height of the rectangle on a particular sub-interval is the value of f at the midpoint of that sub-interval. Hence the name midpoint sum.
As with left- and right-hand sums, we can also find a midpoint sum using graphs or tables—as long as there's enough information to find the midpoint of each sub-interval.
Example 1
Use a midpoint sum with 2 sub-intervals to estimate the area between the function f(x) = x2 + 1 and the x-axis on the interval [0, 4]. |
Example 2
Partial values of the function g are given in the table below.
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Example 3
A graph of the function h is shown below. Use a midpoint sum with 3 sub-intervals to estimate the area between the graph of h and the x-axis on [0, 3].
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Exercise 1
Let f(x) = sin(x) + 1. Use a midpoint sum with 2 equal sub-intervals to estimate the area between f and the x-axis on [0, 2π].
so the rectangle has height
so the rectangle has height
Exercise 2
Let g(x) = x3 on the interval [0, 12]. Use a midpoint sum with 4 equal sub-intervals to estimate the area between g and the x-axis on this interval.
Exercise 3
Partial values of the function h are given in the table below. Use a midpoint sum with 3 intervals to estimate the area between h and the x-axis on [-2, 4] or explain why you can't.
Exercise 4
Partial values of the function f(x) are given in the table below. Use a midpoint sum with 4 equal sub-intervals to estimate the area between f and the x-axis on [0, 1], or explain why you can't.
