Question

a) cartesian

- Y Cone, topped by sphere of radius 7 centered at origin, 90° at vertex For the region W shown in the figure above, write the limits of integration for dV in the following coordinates: JW

b)cylindrical

c) spherical

Answer 1

W : is defined as cone, topped by sphere of readius 7, centereeel al- origin, 900 at vertex: z> Szc2+y? anel x2+y2+z2 = 99 l dv in cartesian coordinates is given by na Vse?+y2 < 25529-42-42 and its projection in x-y CS gennex+y+= 49 plane is a citecle of radius of amccannor n n.

* 14 13 © Sway xon X=76620 y=8 sino 2 -Z Y-JAX2Z = Jxzty z in Cylindreical coorekindtas ir given by x2+y2 = 25 549-42-y2 x2+y2= 19 821 cpOtsin?0) = 49 in 7249 osrst - and 728 17 [ Jay-z-y2 - r dz do do 0=0 r=0 = 5x2+y² I now fore e © Iwdv in spherical Coordinales is given by a= e coso sind y = e sine sind x+y2+z2 = (7) ? 23 e6 le²cos2o sino + e²sin 20 sin26 + 2 cosa = 7 :-/ 82 (60320 + sin2e) sin2p+ 2200320 = 4 i e2 sin20 +p?60424b = #32 .. e? 7. P=7 1 [ose<7) aloo its projection in x-y plane is a circle . 058 seal -

... poso= psing sin2o) (caso, s/n2o) a low fore &, 2=5 az ryz = ecoso= rez(sin24) | 1050+ sim20) · 003p = sino to $=N/A But since the one to vertex of the is at 90° i Ø 305 M. 25 TL 7 the cone at orgin s vertex of p²sind de dodo 0=0 D=TA