Mathematical Methods for Physics and Engineering - Matematica.NET

6.3 APPLICATIONS OF MULTIPLE INTEGRALSydsyȳxFigure 6.7 A curve in the xy-plane, which may be rotated about the x-axisto form a surface of revolution.Pappus’ second theorem states that if a plane curve is rotated about a coplanaraxis that does not intersect it then the area of the surface of revolution so generatedis given by the length of the curve L multiplied by the distance moved by itscentroid (see figure 6.7). This may be proved in a similar manner to the firsttheorem by considering the definition of the centroid of a plane curve,ȳ = 1 L∫yds,and noting that the surface area generated is given by∫S = 2πy ds =2πȳL,which is equal to the length of the curve multiplied by the distance moved by itscentroid.◮ A semicircular uniform lamina is freely suspended from one of its corners. Show that itsstraight edge makes an angle of 23.0 ◦ with the vertical.Referring to figure 6.8, the suspended lamina will have its centre of gravity C verticallybelow the suspension point and its straight edge will make an angle θ =tan −1 (d/a) withthe vertical, where 2a is the diameter of the semicircle and d is the distance of its centreof mass from the diameter.Since rotating the lamina about the diameter generates a sphere of volume 4 3 πa3 , Pappus’first theorem requires that43 πa3 =2πd × 1 2 πa2 .Hence d = 4a4and θ =tan−13π 3π =23.0◦ . ◭197