CK12 Calculus - Single Variable
We can also use this substitutionmethodto evaluatedefiniteintegrals.If we attachlimitsof integrationto our first example,we coul ...
Use integrationby partsmethodto compute Solution: We note that our othersubstitutionmethodis not applicablehere.But our integrat ...
5. 6. 7. 8. 9. 10. Answers ...
10. NumericalIntegration LearningObjectives Use the TrapezoidalRule to solveproblems Estimateerrorsfor the TrapezoidalRule Use ...
The area of a trapezoidis , where and are the lengthsof the parallelsidesand is the height.In our trapezoidsthe heightis and and ...
Of course,this estimateis not nearlyas accurateas we wouldlike. For functionssuchas we can easily find an antiderivative with wh ...
Hencewe musttake to achievethe desiredaccuracy. Fromthe last example,we see one of the weaknessesof the TrapezoidalRule—itis not ...
Suppose for Thenthe errorestimateis givenby Example3: a. Use Simpson’s Rule to approximate with. Solution: We find This turnsout ...
We find that Hencewe musttake to achievethe desiredaccuracy. TechnologyNote:Estimatinga DefiniteIntegralwith a TI-83/84Calculato ...
We usedSimpson’s Rule to solveproblems. We estimatedErrorsfor Simpson’s Rule. ReviewQuestion Use the TrapezoidalRule to appro ...
3. 4. Take Take ...
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5. Applicationsof DefiniteIntegrals............................................................................................. ...
Figure1b Figure1c Therefore,as the graphsshow, it makessenseto say that [Areaunder (Fig. 1a)] [Areaunder (Fig. 1b)] [Areabetween ...
This relationis validas long as the two functionsare continuousand the upperfunction on the interval The AreaBetweenTwo Curves(W ...
and then solvefor from whichwe get and So the upperand lowerboundariesintersectat points and As you can see fromthe graph, and h ...
Figure 3 The AreaBetweenTwo Curves(With respectto the y-axis) If w and v are two continuousfunctionson the interval and for all ...
Figure 4 As you can see from Figure4, the left boundaryis and the right boundaryis The region extendsover the interval However, ...
on the interval integratewith respectto Find the area enclosedby and If the area enclosedby the two functions an ...
Volumes LearningObjectives Learnthe basicconceptsof volumeand how to computeit with a givencross-section Learnhow to computevol ...
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