Internet Resources for the Calculus Student - Topics in Calculus

The two most fundamental examples are the approximation of areas inside curved boundaries by sums of rectangles and triangles, and of the total distance covered in a trip by a sum of contributions from small time intervals on each of which the velocity is approximately constant.

Any quantity computed as a limit of such sums is called a Definite Integral.

The result of considering how such a quantity varies as we change one of the endpoints (eg to find the distance travelled up to time T for a moving object, or the 'area so far' under a function graph from a fixed left end starting point to a variable right end x) is called an Indefinite Integral.

And the connection between Indefinite Integrals and Antiderivatives is important enough to be known as the Fundamental Theorem of Calculus