6.3 APPLICATIONS OF MULTIPLE INTEGRALS
yyds̄yxFigure 6.7 A curve in thexy-plane, which may be rotated about thex-axis
to form a surface of revolution.Pappus’ second theoremstates that if a plane curve is rotated about a coplanaraxis that does not intersect it then the area of thesurface of revolutionso generated
is given by the length of the curveLmultiplied by the distance moved by its
centroid (see figure 6.7). This may be proved in a similar manner to the first
theorem by considering the definition of the centroid of a plane curve,
̄y=1
L∫
yds,and noting that the surface area generated is given by
S=∫
2 πy ds=2π ̄yL,which is equal to the length of the curve multiplied by the distance moved by its
centroid.
A semicircular uniform lamina is freely suspended from one of its corners. Show that its
straight edge makes an angle of 23. 0 ◦with the vertical.Referring to figure 6.8, the suspended lamina will have its centre of gravityCvertically
below the suspension point and its straight edge will make an angleθ=tan−^1 (d/a)with
the vertical, where 2ais the diameter of the semicircle anddis the distance of its centre
of mass from the diameter.
Since rotating the lamina about the diameter generates a sphere of volume^43 πa^3 , Pappus’
first theorem requires that
4
3 πa(^3) =2πd× 1
2 πa
(^2).
Henced=^43 πaandθ=tan−^134 π=23. 0 ◦.