. An elementary treatise on the differential and integral calculus. 5 For the surface of revolution of the whole curve aboutthe initial line, we have re and 0 for the limits of 6, there-fore we have 2nr sin 6 ds (W 8= n = 4:7Ta2 / (1 -f cos 6) cos - sin 0 f*n 6 6= 16r:a2 / cos4 - sm - dB 201. Any Curved Surfaces.—Double Integration.— Let (x, y, z) and (x -f f£r, ?/ + dy, z -f t7z) be two consecu-tive points p and q on the sur-face. Through p let planes bedrawn parallel to the two planesxz and yz; also through q lettwo other planes be drawn par-allel respectively to the planes will
. An elementary treatise on the differential and integral calculus. 5 For the surface of revolution of the whole curve aboutthe initial line, we have re and 0 for the limits of 6, there-fore we have 2nr sin 6 ds (W 8= n = 4:7Ta2 / (1 -f cos 6) cos - sin 0 f*n 6 6= 16r:a2 / cos4 - sm - dB 201. Any Curved Surfaces.—Double Integration.— Let (x, y, z) and (x -f f£r, ?/ + dy, z -f t7z) be two consecu-tive points p and q on the sur-face. Through p let planes bedrawn parallel to the two planesxz and yz; also through q lettwo other planes be drawn par-allel respectively to the planes will intercept aninfinitesimal element pq of thecurved surface, and the projec-tion of this element on theplane of xy will be the infini- ytesimal rectangle PQ, which = dx dy. Let S represent the required area of the whole surface,and dS the area of -the infinitesimal element pq, anddenote by a, (3, y, the direction angles* of the normal atp (x, y, z). Then, since the projection of dS on theplane of xy is the rectangle PQ = dx dy, we have by , Art. 168,. dx dy = dS cos y. (i) * See Anal. Geom., Art. 170. 376 SURFACE OF A SPHERE. Similarly, if dS is projected on the planes yz and zx,we have dy dz = dS cos a ; (2) dz dx = dS cos j8. (3) Squaring (1), (2) and (3), and adding, and extractingthe square root, we have dS = (dx2dy2 + dy2dz2 + dz2dx2)% (since cos2 a + cos2 (3 -f- cos2 y = 1, Anal. Geom., Art. 170). .-. 8 = ff{dx2dy2 + djyW + dz2dx2)\ r ri, dz2 dz2\h 1 _ the limits of the integration depending upon the portionof the surface considered. 202. The Surface of the Eighth Part of a Sphere.— Let the surface represented in Fig. 56 be that of theoctant of a sphere ; then 0 being its centre, its equation is x* + y* + 22 _ a\ Hence, dz _ x dz y dx ~ z dy ~~ z ••• -//(i + J + J)*** /» /> adxdy J J Va2 — x2—y2 Now since pq is the element of the surface, the effectof a ^-integration, x being constant, will be to sum upall the elements similar to pq from H to
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