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Introduction to Definite Integrals - At A Glance:

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. Therefore 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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