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 lefthand sum or righthand sum. For a midpoint sum, the height of the rectangle on a particular subinterval is the value of f at the midpoint of that subinterval. Therefore the name midpoint sum.
As with left and righthand 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 subinterval.
Example 1
Use a midpoint sum with 2 subintervals to estimate the area between the function f (x) = x^{2} + 1 and the xaxis on the interval [0,4]. 
Example 2
Partial values of the function g are given in the table below.

Example 3
A graph of the function h is shown below. Use a midpoint sum with 3 subintervals to estimate the area between the graph of h and the xaxis on [0,3]. 
Exercise 1
Let f (x) = sin x + 1. Use a midpoint sum with 2 equal subintervals to estimate the area between f and the xaxis on [0,2π].
Exercise 2
Let g (x) = x^{3} on the interval [0,12]. Use a midpoint sum with 4 equal subintervals to estimate the area between g and the xaxis 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 xaxis 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 subintervals to estimate the area between f and the xaxis on [0,1], or explain why you can't.