Question

Problem 7. Find the center of mass of the solid bounded by a = yº and the planes = 2, z = 0, and x = 1 if the density is p(x, y, z) = k € R is constant.

Answer 1

Z Z=x plome Z=1 sy &x=y² x=1 x The diagroam of the solid is shown above. Given the den density Plary, z)=k. Total mars of the solid is, 1 x M- SS S Play,z) dx dy dz 2=0 Z=0 y=-v dz S X X 11 K da S S dy X=0 z=0 - vx 1 x da S wa dz X=0 1 X ak sie dae s dz 250

1 7 m = 2K S asa da 0 1 ak[x012] 4K 5 7 = 4K How the x-coordinate of the center of mars 18 x xp (ary,z) da dy dz x=0 =0 y = - 5x M ABS xda s dx s day M - M) S x ada yo x x 252 0 1 ak m. S 25/2 dx 2K M [x42]. - 4* 4K 7 x 5 4K 7 x =

the کم coordinate of the centers of mass is, X +2 Cal s slyplary,z) doe dy dz x=0 Z=0 y = -x M √x x سے da So y dy N=0 2-0 Y = - da 11 da Sdz TO 0 = 0 7 g = z- coordinale of the centers of mass is, The X z S SZP(x, y, z) duchy dz 2=0 7=0 y = -√2 M) ✓a 2 YE s dx z dz s dy X=0 Z=0 y = - 5x 317 da 72 2 27 0 xle 5/2 da m x ak OK IM x ak FM

ㅋ 돈 5 14 4K 미드 So, the centers of mass of the solid is, () = ( 들)이 5 14 Answers