Properties of Definite Integral

Identical Integration Limits

If the upper and lower limits of a definite integral are the same, the result is zero.

This makes a lot of sense intuitively. Since a definite integral calculates the area under a curve over an interval, if the interval has no width (from $a$ to $a$), then there is no area to calculate. It's like trying to find the area of a straight line, which is of course zero.

Reversing the Integration Limits

When we swap the lower and upper limits of an integral, the result is the negative of the original integral's value.

A simple analogy is measuring displacement. The distance from point A to B is the same as the distance from B to A, but the direction is opposite. The negative sign here represents the opposite "direction" in the context of integration.

Constant Multiple Rule

Just as with indefinite integrals, a constant can be factored out of the integral to simplify the calculation.

If we have a function that is "scaled up" or "scaled down" by a constant factor $k$, we can calculate its base area first ($\int f(x)dx$) and then multiply the result by that factor $k$.

Sum and Difference Rule

The integral of a sum or difference of two functions is equal to the sum or difference of their individual integrals.

This property allows us to break down the integral of a complex function into several simpler integrals. We can calculate the area under $f(x)$ and $g(x)$ separately, and then add or subtract them.

Interval Addition Property

An integration interval can be split into several sub-intervals. The total area over the large interval is equal to the sum of the areas of its constituent sub-intervals.

This property holds regardless of the order of $a$, $b$, and $c$. It's like saying that the journey from city A to city C is the same as the journey from A to B plus the journey from B to C. This is a very useful tool for handling functions that are defined differently on different intervals.