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Definite Integrals

Definite Integrals

At a Glance - Midpoint Sum

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.

  • Use a midpoint sum with three sub-intervals to estimate the area between the graph of g and the x-axis on [0, 12].
  • Could we use this table to take a midpoint sum with 4 equal sub-intervals? Why or why not?
  • Could we use this table to take a midpoint sum with 6 equal sub-intervals? Why or why not?

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].


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].


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.


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