A-14 Answers to Odd-Numbered Problems 59 2r 61 1,' 2ry(2 - &)dy = 63 3re 65 Height 1; $: 2rz dz = ra2; cylinder 67 Length of hole is 2d- = 2, so b2 - a2 = 1 and volume is !f 69 F; T(?); F; T Section 8.2 Length of a Plane Curve (page 324) Graphs are flat toward (1,O) then steep up to (1,l); limiting length is 2 ~=\/36sin23t+36cos23t=6 23J,'ady=& 1 J!, J-dy = 3(e'+ e-Y)dy = $(ev - e-')]L1 = e - - e Using x = cosh y this is dy = 1 cosh y dy = sinh y]kl = 2 sinh 1 Ellipse; two y's for the same z 29 Carpet length 2 # straight distance */Z (dd2 = (d~)~ + + (d~)~; ds = \/(%)l + (%)a + (%)2dt; = \/sin2 t + cos2 t + ldt = hdt; 2a4; curve = helix, shadow = circle L = I,' t/TTZ?dz; Jt dGSdz = JGG 2du = 2L; stretch xy plane by 2 (y = x2 becomes : = Section 8.3 Area of a Surface of Revolution (page 327) 1 J" 2rfiJ-dx = 1: 2rdZ+!dx = 3 2 2r(7x)mdx = 14s- 1 5 J',
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