Question

For the lamina that occupies the region D bounded by the curves x = y2 – 2 and x = 2y + 6, and has a density function: p(x, y) = y + 4, find: a) the mass of the lamina; b) the moments of the lamina about x-axis and y-axis; c) the coordinates of the center of mass of the lamina.

Answer 1

Ans Given that, the gregion '' bounded by the Curvece x=9²2 and o density function p(x, y) = y +4 Nee , 1 -2 =h=xt Hi X=2y +ó . and y=x+2 (y-o) = (x-(-2)] Vertex Chik) = (-2,0) x=24+6 x-2y=6 1+ 41 find the point of Interaction from 0 € dy+6= y 2 g² dy=8=0 y = 4ytay-8=0 -

y (7-4) + Cy- 4) =0 (y +2) Cy-4)=0 1 Ysod , 4 (a) Made of the Lamina - M=SS pen, og da timitę :- x - y =2 to 2y +6 y 2 to 4 - 1 4 plyto = f I (4+4) dx dy -2 1-2 - !* (4+) C2y+6=g'se) dy ja Cy+4) Cay+s-y) dy I* (ay?+ 8y = 9P+ 8y +32–445 dy 30-

Le fay?7864 - yº+3x) dy 5 ( 64 - 2y = y + 32) dy - 326494+1(39-(3CJ - 3°(4+2)+8 (16-4) - ſ (64+8) - $60056-) = 192+96-48-60 : masa(4) = 180 (6) Momenta of Lamina about x-axis Mx = 18 9 px, g) da - f g (y + Ajokedy • J cự+ + y)(1 +8 -9*244 ! (24248yty*+8y7 say-49)ing XX

(109= yº-y++38y) dy = 54 (199+1( 924092 Cr - 16C16 –+) + "(64+8) - (256-K) (1056) = 192 +384 -120 - 1056 3 460 - 1056 Moments of Lamina about grauis. на « P(x, y) 44 - +4) de dy 4 Qyto * * $(4+4) [yoy" gangej dy

you! - $ 1.*C4++) ( 44°36+244-44799–47 dy * J*(4+4) (54 +244789?y*dy SC34y +244°489°-45 +128 +964 + B 32 g 2_444 ] dy *J*(128 +1284 +60348yº- +91-99) dy $(128649.71286 2*760(%]*78(900* -419%94-6%2:7 (768 +768 +1440 +480 - 4974-672] 2.4 768 + 8 +1440+4 80 . (e) center of Mass 18 - 48980 - 409775 5 X 180

1244 58180 a center of Mass = (400/45 , 3 1/2as). Thank you.