CALCULUSHigher Concepts of MathematicsThis type of expression is called a summation. A summation indicates the sum of a series ofsimilar quantities. The upper case Greek letter Sigma, , is used to indicate a summation.Generalized subscripts are used to simplify writing summations. For example, the summationgiven in Equation 5-10 would be written in the following manner:(5-11)S 3i1viD tiThe number below the summation sign indicates the value of iin the first term of thesummation; the number above the summation sign indicates the value of i in the last term of thesummation.The summation that results from dividing the time interval into three smaller intervals, as shownin Figure 7, only approximates the distance traveled. However, if the time interval is dividedinto incremental intervals, an exact answer can be obtained. When this is done, the distancetraveled would be written as a summation with an indefinite number of terms.(5-12)S i1viD tiThis expression defines an integral. The symbol for an integral is an elongated "s" . Usingan integral, Equation 5-12 would be written in the following manner:(5-13)S tBtAv dtThis expression is read as S equals the integral of v dt from t = tAto t= tB. The numbers belowand above the integral sign are the limits of the integral. The limits of an integral indicate thevalues at which the summation process, indicated by the integral, begins and ends.As with differentials and derivatives, one of the most important parts of understanding integralsis having a physical interpretation of their meaning. For example, when a relationship is writtenas an integral, the physical meaning, in terms of a summation, should be readily understood.In the previous example, the distance traveled between tA and tB was approximated by equation5-10. Equation 5-13 represents the exact distance traveled and also represents the exact areaunder the curve on figure 7 between tA and tB.Examples:1.Give the physical interpretation of the following equation relatingthe work, W , done when a force, F, moves a body from positionx1to x2.MA-05 Page 42Rev. 0
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