Definite vs. Indefinite: The difference between the two types of integrals is that with Definite integrals, you can find the area underneath a graph that may also be bounded by curves and lines. With those types of integrals, you are also given upper and lower bounds. With Indefinite integrals, you are not given any bounds. Those types of integrals are worked out for the sake of working them out and not solving them after antidifferentiation.

Definite Integrals

Definition. Let f be a function which is continuous on the closed interval [a, b]. The definite integral of f from a to b is defined to be the limit where

Theorem. If f (x) and g(x) are defined and continuous on [a, b], except maybe at a finite number of points, then we have the following linearity principle for the integral:

(i)

f (x) + g(x)dx = f (x) dx + g(x) dx;

(ii)

f (x) dx = f (x) dx, for any arbitrary number .

The next results are very useful in many problems.

Theorem. If f (x) is defined and continuous on [a, b], except maybe at a finite number of points, then we have

(i)

f (x) dx = 0;

(ii)

f (x) dx = f (x) dx + f (x) dx;

(iii)

f (x) dx = - f (x) dx;

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^^A great animation for Integral Funtions^^

As an example, note that since

it follows that

These examples focus upon a specific type of antiderivative, which is the Indefinite kind. In this case, no bounds are set for the function being antidifferentiated.