CK12 Calculus - Single Variable

The FundamentalTheoremof Calculus

LearningObjectives

• Use the FundamentalTheoremof Calculusto evaluatedefiniteintegrals
  Introduction
  In the Lessonon EvaluatingDefiniteIntegrals,we evaluateddefiniteintegralsusingantiderivatives.This
  processwas muchmoreefficientthan usingthe limit definition.In this lessonwe will statethe Fundamental
  Theoremof Calculusand continueto workon methodsfor computingdefiniteintegrals.
  FundamentalTheoremof Calculus:

Let be continuouson the closedinterval

1. If function is definedby on , then on


2. If is any antiderivativeof on then

We first note that we havealreadyprovenpart 2 as Theorem4.1. The proofof part 1 appearsat the end of
this lesson.
Thinkaboutthis Theorem.Two of the majorunsolvedproblemsin scienceand mathematicsturnedout
to be solvedby calculuswhichwas inventedin the seventeenthcentury. Theseare the ancientproblems:

1. Find the areasdefinedby curves,suchas circlesor parabolas.


2. Determinean instantaneousrate of changeor the slopeof a curveat a point.
   With the discoveryof calculus,scienceand mathematicstook hugeleaps,and we can tracethe advances
   of the spaceage directlyto this Theorem.
   Let’s continueto developour strategiesfor computingdefiniteintegrals.We will illustratehow to solvethe
   problemof findingthe area boundedby two or morecurves.
   Example1:

Find the area betweenthe curvesof and
Solution:
We first observethat thereare no limitsof integrationexplicitlystatedhere.Hencewe needto find the limits
by analyzingthe graphof the functions.