CK12 Calculus - Single Variable

We needto use the formulaagainto solvethe integral : If and are both positiveintegers,then an integralof the form can be evaluat ...

Example9: Evaluate. Solution: Here is odd.We followthe thirdprocedurein the table.Makethe substitution, and Our integralbecomes ...

5. 6. 7. 8. Graphand thenfind the volumeof the solidthat resultswhenthe regionenclosedby and is revolvedaroundthe -axis. a. Pro ...

8. 9. TrigonometricSubstitutions LearningObjectives A studentwill be able to: Computeby handthe integralsof a wide varietyof fu ...

In the secondcolumnare listedthe mostcommonsubstitutions.Theycomefrom the referenceright triangles, as shownin the figurebelow. ...

since so that Example2: Evaluate. Solution: Again,we wantto first to eliminatethe radical.Consultthe tableaboveand substitute. T ...

Example3: Evaluate. Solution: Fromthe tableabove,let then Substitutinginto the integral, But since Since Lookingat the triangles ...

ReviewQuestions Evaluatethe integrals. (Hint:First use -substitution,letting ) Graphand then ...

4. 5. 6. 7. 8. 9. Surfacearea is ImproperIntegrals LearningObjectives A studentwill be able to: Computeby handthe integralsof ...

If the integrationof the improperintegralexists,then we say that itconverges. But if the limit of integration fails to exist,the ...

Thusthe integraldiverges. Example2: Evaluate. Solution: Thusthe integrationconvergesto Example3: Evaluate. Solution: Whatwe need ...

Evaluatingthe secondintegralon the right, Addingthe two results, Remark:In the previousexample,we split the integralat However, ...

We next evaluateeachimproperintegral.Integratingthe first integralon the right handside, The integraldivergesbecause and are not ...

Fromthe figureabove,the area of the regionto be revolvedis givenby. Thusthe volumeof the solid is As you can see, we needto inte ...

At this stage,we take the limit as approachesinfinity. Noticethat the whenyou substituteinfinityinto the function,the denominato ...

Similarly, Thusthe integralconvergesto ReviewQuestions Determinewhetherthe followingintegralsare improper. If so, explainwhy. ...

e. Evaluatethe integralor statethat it diverges. 3. 4. 5. 6. 7. The regionbetweenthe -axisand the curve for is revolvedabou ...

4. divergent divergent a. b. Homework Evaluatethe followingintegrals. Graph and find the volu ...

The integralconvergesfor all a. Find b. Provethat , for all. c. Provethat Referto the GammaFunctiondefinedin the previousexerci ...

a. Hint: Let b. Hint: OrdinaryDifferentialEquations Generaland ParticularSolutions Differentialequationsappearin almostevery ...