. Applied calculus; principles and applications . to OX and represented in length by NNi, althoughit is not the same as Ax = dx but is really longer. Example 1. — To find the lateral surface of the cone ofArt. 155: by (3), S = 2^ I yds = 2Trl -^ sds, where s = -y = OPf n Si-= 2 TT 7 pr = -wal, where I = OPh, an element. 6 z Jo Again, J^Sft /•2/=a I /I \ I yds = 2Tr I y-dy, since ds = d(-y]=-dy^0 Jo ct \a / a limit or end value being changed from I to a. Example 2. — To find the surface of the paraboloid ofArt. 156: J^l 1 1 2/ [1 + 64 y^]dy, from 2/^ = 7 x,0 4 .[l + (64,^)i]J = ^ ((65)1-1) 27r1
. Applied calculus; principles and applications . to OX and represented in length by NNi, althoughit is not the same as Ax = dx but is really longer. Example 1. — To find the lateral surface of the cone ofArt. 155: by (3), S = 2^ I yds = 2Trl -^ sds, where s = -y = OPf n Si-= 2 TT 7 pr = -wal, where I = OPh, an element. 6 z Jo Again, J^Sft /•2/=a I /I \ I yds = 2Tr I y-dy, since ds = d(-y]=-dy^0 Jo ct \a / a limit or end value being changed from I to a. Example 2. — To find the surface of the paraboloid ofArt. 156: J^l 1 1 2/ [1 + 64 y^]dy, from 2/^ = 7 x,0 4 .[l + (64,^)i]J = ^ ((65)1-1) 27r1281 0^ (65 V65 — 1) square units. 274 INTEGRAL CALCULUS Example 3. — To find the surface of the sphere of Art. 155,or any part of it, as a zone. For a change take origin at A on the circumference,making y = a/2 ax — x^ and 2 iry the curve P bounding thesection A^; then by (2) or (3), Pds = 2t I yds = 2t I adx 0 Jq Jxo (where y ds = a dx, from similar triangles, OMP and PDT) ]x I2a = 2 7ra (o; — Xo) or 2 i^ax — 4 Tra^.xo Jo Ddx T. Drawing PT = ds from P parallel to a;-axis, 2 iry ds is thelateral surface of the cylinder PT\ which is equal in areato that of the cylinder DT\ which is 2 ira volume is again, with origin at A, by (4), yidx = ir j^ {2ax-x)dx = ir\^-^\ = 7ra^ Example 4. — To find the lateral surface of a quadrangularpyramid. Let Ph = perimeter of base and I = OPh = slantheight. Let PMN be the position of the generating perim-eter P when s = OP. Since P and Ph are similar, SURFACE AND VOLUME OF ANY FRUSTUM 275 PPh OPOPh s hence, P = -^ s, in (2); ^ = Jo ^^^ = TJo^^^ = T2jo=2- that is, the convex surface of any pjramid or cone (Ex. 1)is measured by half the product of perimeter of base andslant
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