Question

Bonus. (8 pts) Many of the definitions we used for two-dimensional mass and moments can be extended to three dimensions rather easily. For example, mass = ///. P18,9, 2) dv would represent the mass of the solid Q where p(x, y, z) is the density at any point (x, y, z). Find the mass of the solid bounded laterally by the cylinder zº + y2 = 2x and bounded above and below by the cone 2? = 2² + y2. Here, the density of the solid is proportional to the distance away from the z-axis.

Answer 1

Solution: Mass 乙 SSS, PC,y,z) av where a: bounded above cylinder vry? = 24 and core 2 = ury also p(x, 4. 2) is density, Roponitional to the distance away from z-axis. ² = uity? IZ ne=ht clearly z=o to z=rity 2 Y also we are given 2 f(u, y, z) a z Х. zx=12'R'NIJE 22 Mass= nity dydu <-combat SSS kz az uydu - SKLO HASS Getty Jaydn Rojection zzo on X-y Plane

let a= ricore, y =usine Bojection on X-y Plane (seity =1 5 isa ircle kty=22 » r = goose Mass=k quoce 633 ndade G=-* rzo 1 96 goose de X d 6 T k od sa cokode * *2S coorde 33k 6-1)(6-3) 6-5) (uning hallix formala 32 داني Mass /V(بر) 37