CK12 Calculus - Single Variable

3. 
   

   ; (notethat we haveincludedareasunderthe x-axisas negativevalues.)

   
   


4. 
   

   ;

   
   


5. 
   

   ;

   
   


6. 
   


7. 
   


8. 
   


9. 
   

   Yes, since is continuouson

   
   


10. 
    

    No, since ;

    
    

DefiniteIntegrals

LearningObjectives

• Use RiemannSumsto approximateareasundercurves


• Evaluatedefiniteintegralsas limitsof RiemannSums
  Introduction
  In the LessonThe AreaProblemwe definedthe area undera curvein termsof a limit of sums.

where

and were examplesofRiemannSums. In general,RiemannSumsare of form
whereeach is the valuewe use to find the lengthof the rectanglein the sub-interval.For example,
we usedthe maximumfunctionvaluein eachsub-intervalto find the uppersumsand the minimumfunction
in eachsub-intervalto find the lowersums.But sincethe functionis continuous,we couldhaveusedany