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f Math 206A Quiz 07 page 1 Monday 12/06/2010 | Name supgartel coe 1. Find a parameterization f and corresponding region R for the helicoid shown. Note the helicoid wraps around the z-axis three times, and the points on the outer helix boundary are 2 units from the z-axis; finally, the bottom is 1 unit below the x axis and the top is 2 units above it. tike f(t) = (4085, tins, ie there sé [-ar, dr] a télo2] (2 R-[-2, a] « [23 ~~ (0,0,2. 2. Recall one way to parameterize the unit sphere is £(s, t) = (cos (s) cos (£) , cos (s) sin (¢) , sin (s)), for (s,¢) in the rectangle R = [—1/2,1/2] x [0,2z]. Find a subsetA of R such that the parameterization f restricted to A yields the part of the sphere shaded in below. The lines connecting the poles are called meridians, or lines of longitude. They are 9 degrees apart on this figure. The bold meridian passes through (1,0,0). Also shown is the “equator”, which is the unit circle in the ry plane. The lines running parallel to it are called “parallels”, or lines of latitude. On this figure they are 6 degrees apart. Give your answer in terms of radians. th shad reyon atte tom 5x6°>30"= Uh Nort. Inthe’ fp JS *b67> ys? = My. at sho toves fom -5#9° fo 549% 2 -18° b 45? Fan “wet *& east Hovewr, ve pce Comtpondh Valen tn DO f Orr Mk -45° fo 15%, or “Ty become 25 bln feral vith 0% Tus ts e(%, 4) £ €[ 27, aJulo, ta] om the ee dekh ic [%, Ma] x ([2, an)» a 3. Consider the surface M having the parameterization given by £(s,t) = (s?, #3, s + t) over the region in the st plane bounded by s = 1, s = 2 and the curves t = s* and t = 6 — s. Set up the double integral (with appropriate limits on the integrals) which represents the surface area of M. Simplify the integrand as much as Byes. ? eee ee So een Jr ia x 2t) de db jase s* ARE aes no, a 2s, 0, 1) x (0, 3, (0,38, 1) = (-3, -2s, ost) : Jo || $ # x 7 =f (-3e)te (-25)*-¢ Gob)” = [F444 43s" 4 2 ss ont Hef om a i = [9a de® LH" oft as 4. Let M be as in (3). Let g(x,y, z) =y+-
