Now another function is going to join the party. Assume both f and g are integrable functions.
Also, the integral of a difference is the difference of the integrals:
We've taken whatever weighted area was between g and the x-axis, and stuck that on top of f. If g is negative, in some places we may actually be subtracting area from f. The integral of (f + g) is the integral of f plus the integral of g.
f (x) < g(x) for all x in [a,b]
This is using the last property. We're comparing f to the constant functions m and M. If m ≤ f (x) on [a,b], then we know from the previous property that
Since , we have half the inequality explained.
Similarly, if f ( x ) ≤ M on [a,b] then we know from the previous property that
and we know that .
If f ( x ) < 2x, then
We can find by looking at the graph and getting the area of the triangle:
We conclude that
If and , then
Be Careful: The integral of a product is not necessarily the product of the integrals.