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
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