Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

z

x

y

a

a

a

√

a^2 −z^2

dz

Figure 6.5 The solid hemisphere bounded by the surfacesx^2 +y^2 +z^2 =a^2

and thexy-plane.

A

y

y

x

dA

̄y

Figure 6.6 An areaAin thexy-plane, which may be rotated about thex-axis

to form a volume of revolution.

If a plane area is rotated about an axis that does not intersect it then the solid

so generated is called avolume of revolution.Pappus’ first theoremstates that the

volume of such a solid is given by the plane areaAmultiplied by the distance

moved by its centroid (see figure 6.6). This may be proved by considering the

definition of the centroid of the plane area as the position of the centre of mass

if the density is uniform, so that

̄y=

1

A

∫

ydA.

Now the volume generated by rotating the plane area about thex-axis is given by

V=

∫

2 πy dA=2π ̄yA,

which is the area multiplied by the distance moved by the centroid.