Question

Question #4: 3 pts each] Consider a solid D in the first octant bounded below by z= 14-x'-y? and bounded above by Vis? + y’, for y20. ZE a) Find the intersection of the surfaces. b) Setup the triple integral (without evaluating) in rectangular coordinates. c) Setup the triple integral (without evaluating) in cylindrical coordinates. d) Setup the triple integral (without evaluating) in spherical coordinates

Answer 1

Given a solid in the first octant bounded above by z= 4– x2 - y2 and below by z= vz? +r?, y20. a). intersection 4-? V2+ ye? 4- x2 - y2 = x2 + y2 =x + y2 = 2 =x+y+= (v2), whichis a circle of radius V7with centre (0,0). b). In rectangular coordinates 17 dzdydx c) In cylindrical coordinates III dzdydx= (x,y,z Idz ar de 0 0 0 0 | rdz dr de 0 0 By the relation between cartesian coordinates (x,y,z) and the cylindrical coordinates (r,8,2), we have x=rcoso (1); y=r sin 8 → (2); z=z →(3), where Vz2+ yd is distance of Q(x,y,0) from 0(0,0,0), is the angle made by og on the xy-plane with respect to positive x-axis d). In spherical polar coordinates x,y,z drdo de 0 0 L200 . pa sin Ødrdo de 200 :(upper hemi-sphere)2 = 14- x? - ya =rcosø = 4–ma sinº ø=p2 cosao=4–r? sina ø=r=2. Now (cone) z = 2 + y =>r cos $ = waº sinºp=rcosø=r sin ø=•=* In spherical polar coordinates (r,0,0): By the relation between cartesian coordinates (x,y,z) and thespherical polar coordinates (r, 0,7), we have x=r sin cos 8 →(1); y=r sin Øsin 8 → (2); z=rcos (3), where ris distance of P(x,y,z) from 0(0,0,0), Øis the angle made by OPwith respect to positive z-axis and O is the angle made by the projection of Op on the xy-plane with respect to positive x-axis.