If is continuouson the closedinterval then
where is any antiderivativeof
We sometimesuse the followingshorthandnotationto indicate
The proofof this theoremis includedat the end of this lesson.Theorem4.1 is usuallystatedas a part of
the FundamentalTheoremof Calculus,a theoremthat we will presentin the Lessonon the Fundamental
Theoremof Calculus.For now, the resultprovidesa usefuland efficientway to computedefiniteintegrals.
We needonly find an antiderivativeof the givenfunctionin orderto computeits integralover the closedin-
terval.It also givesus a resultwith whichwe can now stateand provea versionof the MeanValue Theorem
for integrals.But first let’s look at a coupleof examples.
Example1:
Computethe followingdefiniteintegral:
Solution:
Usingthe limit definitionwe foundthat We now can verifythis usingthe theoremas follows:
We first note that is an antiderivativeof Hencewe have
We concludethe lessonby statingthe rulesfor definiteintegrals,mostof whichparallelthe ruleswe stated
for the generalindefiniteintegrals.