CK12 Calculus - Single Variable
; (notethat we haveincludedareasunderthe x-axisas negativevalues.) ; ; Yes, since is continuouson No, since ; ...
pointswithinthe sub-intervalsto find the limit. Hencewe can definethe mostgeneralsituationas follows: Definition: If is continuo ...
Now let’s computethe definiteintegralusingour definitionand also someof our summationformulas. Example2: Use the definitionof th ...
compute. LessonSummary We usedRiemannSumsto approximateareasundercurves. We evaluateddefiniteintegralsas limitsof RiemannSums. ...
Find the total area boundedby the graphof and the -axis,from to to Use your knowledgeof geometryto evaluatethe followingdefinit ...
If is continuouson the closedinterval then where is any antiderivativeof We sometimesuse the followingshorthandnotationto indica ...
where Giventheserulestogetherwith Theorem4.1, we will be able to solvea greatvarietyof definiteintegrals. Example2: Compute Solu ...
Consider We will now employa methodthat will expressthe right side of this equationas a Riemann Sum. In particular, Notethat is ...
FromTheorem4.1 we havethat is an antiderivativeof Hence, and in particular, Hence,by substitutionwe have Notethat Hencewe have a ...
7. Find the averagevalueof over If is continuousand showthat takeson the value at leastonceon the interval Your friendstatestha ...
The FundamentalTheoremof Calculus LearningObjectives Use the FundamentalTheoremof Calculusto evaluatedefiniteintegrals Introduc ...
We observethat the regionsof interestare in the first and third quadrantsfrom to We also observethe symmetryof the graphsaboutth ...
. Example3: Find the area enclosedby the curvesof and Solution: The graphindicatesthe area we needto focuson. Beforeprovidingano ...
Solution: Whilewe couldeasilyintegratethe right side and then differentiate,the FundamentalTheoremenablesus to find the answerve ...
2. Then by our rulesfor definiteintegrals. Then. Hence Since is continuouson and then we can select suchthat is the minimumvalu ...
4. (Hint:Examinethe graphof the functionand dividethe intervalaccordingly.) In problems#5–7use antiderivativesto computethe defi ...
3. 4. 5. 6. ...
7. 8. 9. 10. ...
Integrationby Substitution LearningObjectives Integratecompositefunctions Use changeof variablesto evaluatedefiniteintegrals Us ...
Differentiateboth sidesso Changethe originalintegralin to an integralin : where and Integratewith respectto : Changethe answe ...
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