CK12 Calculus - Single Variable
Figure 6 The volumeof soliddoesnot necessarilyhaveto be circular. It can take any arbitraryshape.One useful way to find the volu ...
If we use the sameargumentto derivea formulato calculatethe area underthe curve,let us increasethe numberof slicesin sucha way t ...
Figure7a ...
Figure7b Solution: Let the -axispassthroughthe apexof the pyramid,as shownin Figure(7a). At any point in the interval the cross- ...
Figure8a ...
Figure8b Supposea function is continuousand non-negativeon the interval and supposethat is the regionbetweenthe curve and the -a ...
Volumesby the Methodof Disks(revolutionaboutthe -axis) Becausethe shapesof the cross-sectionsare circularor look like the shapes ...
Figure9a ...
Figure9b Example3: Derivea formulafor the volumeof the spherewith radius Solution: One way to find the formulais to use the disk ...
This is the standardformulafor the volumeof the sphere. The Methodof Washers To generalizeour results,if f and g are non-negativ ...
Figure10a Figure10b Volumesby the Methodof Washers(revolutionaboutthe -axis) ...
Example4: Find the volumegeneratedwhenthe regionbetweenthe graphs and over the interval is revolvedaboutthe -axis. Solution: Fig ...
Figure11b Fromthe formulaabove, The methodsof disksand washerscan also be used if the regionis revolvedaboutthe -axis.The analog ...
Washers: Example5: Whatis the volumeof the solidgeneratedwhenthe regionenclosedby and is revolvedaboutthe -axis? Solution: Since ...
Figure12a ...
Figure12b VolumeBy CylindricalShells The methodof computingvolumesso far dependeduponcomputingthe cross-sectionalarea of the sol ...
Figure13a Figure13b If we wish to integratewith respectto the -axis,we haveto solvefor in termsof That wouldnot be easy(try it!) ...
Figure 14 Noticehoweverthat is the thicknessof the shelland is the averageradiusof the shell. ...
Thus Replacingthe averageradiuswith a singlevariable and using for the height,we have In generalthe shell’s thicknesswill be or ...
Figure 15 Solution: FromFigure15 we can identifythe limitsof integration: runs from to A horizontalstrip of this region wouldgen ...
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