Mathematical Expression:

This kind of expression is known as a summation. A summation denotes the sum of a series of similar quantities. The upper case Greek letter Sigma, Σ, is used to denotes a summation. Generalized subscripts are used to simplify writing summations. For instance, the summation given in Equation 10 would be written in the following manner:

The  number  below  the  summation  sign  denotes  the  value  of  i  in  the  first  word  of  the summation; the number above the summation sign denotes the value of i in the last term of the summation.

The summation which results from dividing the time interval within three smaller intervals, as shown in Figure, only approximates the distance traveled.   Therefore, if the time interval is divided within incremental intervals, an exact answer can be acquired.   While this is done, the distance traveled would be written as a summation along with an indefinite number of terms.

This expression describes an integral. The symbol for an integral is an elongated "s". Using an integral, Equation 12 would be written in the subsequent manner:

This expression is read as S equals the integral of v dt from t = t_A to t = t_B.  The numbers below & above the integral sign are the limits of the integral.  The limits of an integral denote the values at that the summation process, denotes through the integral, begins and ends.

As with differentials and derivatives, one of the most significant parts of understanding integrals is having a physical interpretation of their meaning.  For instance, while a relationship is written as an integral, the physical meaning, in terms of a summation, should be readily understood. In the previous instance, the distance traveled between t_A  and t_B  was approximated by equation 10.   Equation 13 represents the exact distance traveled and also represents the exact area under the curve on figure among t_A   and t_B .