CK12 Calculus - Single Variable
Note:The alert readerwill havenoticedthat this examplecouldbe workedwith a simplerintegralusing disks.However, the followingexam ...
ReviewQuestions In problems#1–4,find the volumeof the solidgeneratedby revolvingthe regionboundedby the curves aboutthe x-axis. ...
aboutthe -axis. Use the cylindricalshellsmethodto find the volumegeneratedwhenthe regionis boundedby is revolvedaboutthe lin ...
In this sectionwill considerthe problemof findingthe lengthof a planecurve.Formulasfor findingthe arcs of circlesappearedin earl ...
Figure 18 Solution: Since Usingthe formulaabove,we get Using -substitutionby letting , then Substituting,and rememberingto chang ...
ReviewQuestions Find the arc lengthof the curve on Find the arc lengthof the curve on Integrate Find the lengthof the curve ...
The figurebelowshowsa suspensionbridge.The cablehas the shapeof a parabolawith equation The suspensionbridgehas a total lengtho ...
Answers Areaof a Surfaceof Revolution LearningObjectives A studentwill be able to: Learnhow to find the area of ...
Figure 19 Figure 20 ...
Areaof a Surfaceof Revolution If f is a smoothand non-negativefunctionin the interval then the surfacearea generatedby re- volvi ...
Figure 21 The surfacearea S is Using -substitutionby letting ...
Example2: Find the area of the surfacegeneratedby revolvingthe graphof on the interval about the -axis(Figure22). Solution: Figu ...
So we write as. In addition,the intervalon the -axis becomes Thus Simplifying, With the aid of -substitution,let ReviewQuestions ...
Answers ApplicationsfromPhysics,Engineering,and Statistics LearningObjectives A studentwill be able to: ...
Solution: Usingthe formulaabove, Example2: A librariandisplacesa bookfroman uppershelfto a lowerone. If the verticaldistancebetw ...
2.The sandalone.The weightof the sandis decreasingat a constantrate from N to N over the -meterlift. Whenthe bucketis at metersa ...
3.The rope alone.Sincethe weightof the rope is N/m and the heightis meters,the total weightof the rope from the floor to a heigh ...
where is the densityof the fluid and is the accelerationdue to gravity(whichis on Earth).The pressurethen can be writtenas Examp ...
As you can see, it is easyto calculatethe fluid forceon a horizontalsurfacebecauseeachpointon the surfaceis at the samedepth.The ...
Figure 24 Solution: Let the depthof the water. At an arbitrarypoint on the dam,the widthof the dam is m and the depthis m. The w ...
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