. Differential and integral calculus, an introductory course for colleges and engineering schools. 216 MULTIPLE INTEGRALS 329 It is easily proved that a = acos0, and from this it follows thatlim Va= cosfllim^a, or A = Acosd. Q. E. D. Now let the equation of the curved surface be f(x, y, z) = 0,and let S be a portion of this surface bounded by a closed contour seek a formula for the area of S. Let S be projected into a region S of the xy-plsme. We assumethat the perpendiculars upon the x?/-plane which project S intoS meet S in but one point; that is, weassume z to be a single-valued functi
. Differential and integral calculus, an introductory course for colleges and engineering schools. 216 MULTIPLE INTEGRALS 329 It is easily proved that a = acos0, and from this it follows thatlim Va= cosfllim^a, or A = Acosd. Q. E. D. Now let the equation of the curved surface be f(x, y, z) = 0,and let S be a portion of this surface bounded by a closed contour seek a formula for the area of S. Let S be projected into a region S of the xy-plsme. We assumethat the perpendiculars upon the x?/-plane which project S intoS meet S in but one point; that is, weassume z to be a single-valued function ofx and y throughout S.* Circumscribeabout S a polyhedral surface P, by con-structing tangent planes to S. As thenumber of tangent planes is indefinitelyincreased, the number of faces of P isindefinitely increased, and the area of Papproaches as a limit the area of $.f Rep-resenting the faces of P by APi, AP2, • • • ,it is fairly obvious that. S = lim J) AP. The faces of P project into a network of polygons covering (x, y, z) be the point of contact of the plane of AP with S,and AS the projection of AP in S. Let 6 be the angle betweenthe plane of AP and the xy-plsme. Then by the lemma AS AS = AP cos 6, whence AP = cos d Now 6 is equal to the angle between OZ and the normal to S at(x, y, z); that is, 6 is equal to one of the direction angles of thisnormal. Therefore by Art. 209, cos 9 dz Vr whence AP = --j- A*Sr,dz * Of course z is also assumed to be real and continuous throughout The tangent planes must be so drawn that each face of P has the limit 0. 330 INTEGRAL CALCULUS §216 where MSMfHI) a/7? Hence S = lim V AP = lim V ~ AS, 17 IF df and finally, -s S dz x=S |/=/3(x)
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