These properties require a little more explanation. We're still assuming f is an integrable function.
- Let c be a constant. Then
As an example, let f = x on [0, b] and let c = 3.
When we stretch f by 3 we don't change the base of the triangle, but we do stretch the height by 3. We also multiply the area by 3.
When we rewrite
we say we're "pulling out the constant" or "pulling the constant out of the integral."
- We can switch the limits of integration if we also switch the sign:
We've been focusing on integrals from a to b where a < b. If f is non-negative then as we accumulate area from left to right, we weight the area positively. As we accumulate area from right to left, we weight it negatively.
The integral of f on [a, b] is the weighted area between f and the x-axis on [a, b]. The integral of f on [b,c] is the weighted area between f and the x-axis on [b, c]. When we add these weighted areas, we get the weighted area between f and the x-axis on [a, c], which is the integral of f on [a, c]. When a,b,c don't fall nicely in order, this is still true.
- If f (x) is an even function then
If f is even then it's reflected across the y-axis.This means whatever area is on one side of the graph is also on the other side of the graph. The weighted area between f and the x-axis on [-a, a] is then double the weighted area between f and the x-axis on [0, a].
- If f is an odd function then
You can see that the area between f and the horizontal axis is the same on both sides, but on one side the area is above the x-axis and on the other it's below the x-axis. Odd functions will always work this way. If we integrate from -a to a the weighted areas above and below the x-axis will cancel each other out.