Elements of analysis as applied to the mechanics of engineering and machinery . ^^^ Mx\ C now put the area of such an element of surface, ^ -J. J/Y=(y-f I dy) dx = ydx, g^=(5^ + ^^«^ the area of the entire surface F may be found by integrating thedifferential ydx, thus putting F =f example, for a parabola with the parameter p, we have y^ =px,and hence the surface of the same: / 3 V~p^dx = Vpj xdx = —-3--- = I ^ V px^ F xy. The parabolic surface AB G is., therefore, two-thirds of the rec-tano-le AGB1) which encloses it. Art. 29.] ELEMENTS OF ANALYSIS. 43 This formula is also applicable
Elements of analysis as applied to the mechanics of engineering and machinery . ^^^ Mx\ C now put the area of such an element of surface, ^ -J. J/Y=(y-f I dy) dx = ydx, g^=(5^ + ^^«^ the area of the entire surface F may be found by integrating thedifferential ydx, thus putting F =f example, for a parabola with the parameter p, we have y^ =px,and hence the surface of the same: / 3 V~p^dx = Vpj xdx = —-3--- = I ^ V px^ F xy. The parabolic surface AB G is., therefore, two-thirds of the rec-tano-le AGB1) which encloses it. Art. 29.] ELEMENTS OF ANALYSIS. 43 This formula is also applicable to oblique angled co-ordinatesintersecting each other at an angle XA Y = a] for instance, to thesurface AB G^ Fig. 36, if, instead of B G = y, the normal distanceB N = y sin. a be substituted; we have here, therefore, F ^= sin. a f ydx. For the parabolic surface, for example, when the axis of abscissas A X forms a diameter, and the axis of ordinates AY^Si tangent, of pxthe parabola, and therefore, y^ = p^ x = sin a 2 5 there results JF =z ^ xy sin. «, 1. e.: surface AB G. I parallelogram AGBD. Fig. 37.
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