CK12 Calculus - Single Variable

If we use the sameargumentto derivea formulato calculatethe area underthe curve,let us increasethe
numberof slicesin sucha way that. In this case,the slicesbecomethinnerand thinnerand,
as a result,our approximationwill get betterand better. That is,

Noticethat the right-handside is just the definitionof the definiteintegral.Thus

The VolumeFormula(Cross-sectionperpendicularto the -axis)

Let be a solidboundedby two parallelplanesperpendicularto the -axisat and If

eachof the cross-sectionalareasin are perpendicularto the x- axis,then the volumeof the solidis

givenby

where is the area of a crosssectionat the valueof x on the x-axis.

The VolumeFormula(Cross-sectionperpendicularto the -axis)

Let S be a solid boundedby two parallelplanesperpendicularto the -axisat and If each

of the cross-sectionalareasin are perpendicularto the -axis,then the volumeof the solid is given

by

where is the area of a crosssectionat the valueof on the -axis.

Example1:
Derivea formulafor the volumeof a pyramidwhosebaseis a squareof sides and whoseheight(altitude)
is