. Applied calculus; principles and applications . the two cylinders be x^ -\- z^ = a^ and y^ -\- z^ = a^;then A, = LMPN = xy = a^ - z^, and 8 pA. dz = Sr (a2 - ^2) dz = S \ah - ^1 16 3 The total volume common, being 8 times Z — OACB, is ^3^- aP,Example 8. A dome has the shape of the figure of Ex. 7, find the area of the curved surface. The surface ZBC is equal in area to the surface ZAC, and is one-eighth part of the surface of the dome, which surface is the upper half of the surface of the common volume of Hence the surface of the dome of eight equal parts is given by SURFACE AND VOLUME
. Applied calculus; principles and applications . the two cylinders be x^ -\- z^ = a^ and y^ -\- z^ = a^;then A, = LMPN = xy = a^ - z^, and 8 pA. dz = Sr (a2 - ^2) dz = S \ah - ^1 16 3 The total volume common, being 8 times Z — OACB, is ^3^- aP,Example 8. A dome has the shape of the figure of Ex. 7, find the area of the curved surface. The surface ZBC is equal in area to the surface ZAC, and is one-eighth part of the surface of the dome, which surface is the upper half of the surface of the common volume of Hence the surface of the dome of eight equal parts is given by SURFACE AND VOLUME OF ANY FRUSTUM 279 S = 8ZBC = 8 r^Pds = 8 r^NPds -dz = 8a I dz = Sa^.0 y Jo The result shows that each of the curved surfaces of the sohdZ — OACB is equal in area toits base OACB; the surface ofthe dome being just twice thatof its base. Note. — Another determina- ^tion of the area of ZBC maybe made by developing thecurved surface upon a planeand finding the area as a planearea. Thus, developing ZBCas the plane area ZBC, with B as origin;.
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