We then summedthe areasof the rectanglesas follows:
and
We call this theuppersumsinceit is basedon takingthe maximumvalueof the functionwithineachsub-
interval.We notedthat as we usedmorerectangles,our area approximationbecamemoreaccurate.
We wouldlike to formalizethis approachfor both upperand lowersums.First we note that thelowersums
of the area of the rectanglesresultsin Our intuitiontells us that
the true area lies somewherebetweenthesetwo sums,or and that we will get closer
to it by usingmoreand morerectanglesin our approximationscheme.
In orderto formalizethe use of sumsto computeareas,we will needsomeadditionalnotationand terminology.
SigmaNotation
In The LessonThe Calculuswe useda notationto indicatethe uppersum whenwe increasedour rectangles
to and foundthat our approximation. The notationwe used to enabled
us to indicatethe sum withoutthe needto writeout all of the individualterms.We will makeuse of this no-
tationas we developmoreformaldefinitionsof the area underthe curve.
Let’s be moreprecisewith the notation.For example,the quantity was foundby summingthe
areasof rectangles.We wantto indicatethis process,and we can do so by providingindicesto
the symbolsusedas follows:
The sigmasymbolwith theseindicestells us how the rectanglesare labeledand how manytermsare in
the sum.
UsefulSummationFormulas