CK12 Calculus - Single Variable

Thus

Replacingthe averageradiuswith a singlevariable and using for the height,we have

In generalthe shell’s thicknesswill be or dependingon the axis of revolution.This discussionleads
to the followingformulasfor rotationaboutan axis. We will then use this formulato computethe volume
of the solid of revolutionthat is generatedby revolvingthe regionaboutthe -axis.

VolumeBy CylindricalShellaboutthe y-Axis

Suppose is a continuousfunctionin the interval and the region is boundedaboveby

and belowby the -axis,and on the sidesby the lines and If is rotatedaroundthe

-axis,then the cylindersare vertical,with and The volumeof the solid is givenby

VolumeBy CylindricalShellaboutthe -Axis

Equivalently, if the volumeis generatedby revolvingthe sameregionaboutthe x-axis,then the cylinders

are horizontalwith

where and The valuesof and are determinedin context,as you will see

in Example6.

Note:Example7 showswhatto do whenthe rotationis not aboutan axis.

Example6:

A solidfigureis createdby rotatingthe region (Figure15) aroundthe -axis. is boundedby the
curve and the lines and Use the shell methodto computethe volumeof the solid.