Section 5.2 , the integral from a to b of f(x) .

For any continuous function, f(x), where is the left hand sum and is the right hand sum for n equal subintervals of [a, b].

The general definition of integrable and the general Riemann sum are in the text. The intervals need not be equal and any function value in a subinterval can be the "height" for the rectangle. If the function is negative in the subinterval, the contribution from that subinterval is negative. We will use functions which are continuous but not necessarily non-negative but will not use general Riemann sums. We will use , , and their average.

Example: Find , , and their average for on [-1, 2] with n=6.

The partition of [-1, 2] has subinterval lengths We start with -1 and add increments of 1/2 until we get to 2.

For we use all but the last, for we use all but the first of these values in the function.

= (1/2)(-1) + (1/2)(-1/8) + (1/2)(0) + (1/2)(1/8) + (1/2)(1) + (1/2)(27/8)

= (1/2)[ -1 + -1/8 + 0 + 1/8 + 1 + 27/8 ]= 27/16

Similarly,

= (1/2)[ -1/8 +0 + 1/8 +1 + 27/8 + 8 ] = 99/16

Properties of :

= the Area Above the x-axis under f minus the Area Below the x-axis above f