Question

D . Problem 4. A lamina lies in the first quadrant and is enclosed by the circle x2 +y2 = 4 and the lines x = 0 and y = 0. The density function of the lamina is equal to p(x, y) = V x2 + y2. Use the double integral formula in polar coordinates, S/ s(8,y)dx= $." \* fcr cos 6,r sin Øyrar] de, Ja [ Ꭱ . to calculate (1) the mass of the lamina, m = SSP(x,y) dĀ, (5 points) (2) the moment of the lamina about the y-axis, M, = SSx p(x, y) dĀ, (5 points) (3) the moment of the lamina about the x-axis, Mg = SSy p(x, y) dA, (5 points) R (4) the coordinates (ū, y) of the center of mass of the lamina, ī=; point) R N

Answer 1

otoco 1 een that lamine lies in the fist quetrant enclord 4 0= &ty=4 and a=0, y=0 for chanque entro polar Coordinates x=raoso, y=rfro 1. rtyar dudy = r drdo The limita an polar coordinates ryo to 2 o- o to ne azo, 6=2 P=0, ß =% Here density function Play) = [artyr = for ar The mass of the lamina, m= J J per, 47 da TULA2 1) no ao a the momentum of laming about g axu My = $) apcauda = S "Jercado) (1) rdrdo - wo do preko = ()* )=6:9]) TI about an axis The momentum of lamine ohne O Mx = Sf 9 p (94) da = 11th / Cr GG) (1) (+) do do for the do Stick = (-60 -(62-680) (-o) 3 ME : center og mom (6,9 ) = 6 ),