Question

If I'm given a table that gives f(x) and am asked to find the average value using the Midpoint rule, don't I just find the average between the f(x) numbers, add them, and then multiply by change in x?

Answer 1

If you have a table of values of f(x), you use the midpoint rule to find the average between each successive pair of values of f(x), then multiply each average by the "width" of the box (the difference between the values of x). Add all of those results, and divide by the change in x from the start of the region to the end.

For example, if you have the table

x f(x)
------
0 3
1 5
2 5
4 11

The midpoint between x = 0 and x = 1 is (3+5)/2=4. (1-0)*4 = 4
The midpoint between x = 1 and x = 2 is (5+5)/2 = 5. (2-1)*5 = 5
The midpoint between x = 2 and x = 4 is (11+5)/2=8. (4-2)*8 = 16

The average is (4+5+16)/(4-0) = 25/4

Here's a graph of the function (in blue) I used to make the table, and the (continuous) average value (in purple). The horizontal line indicates 25/4.