Answer Happy

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is bounded by y=ex,y=0,x=0, and x=1; ρ(x,y)=31y Step 1 The mass can be found by integrating the density function ρ(x,y) over the region, therefore, ∬D​ρ(x,y)dA=∫0ax​∫0ex​ydydx. Step 2 The inner integral is ∫0ex​31ydy=[231y2​]]0x​=231c2x​.  Step 3 Therefore, the mass is m=∫01​231​e2xdx=[∣01​= Step 4 The center of mass is the point (xy​​yˉ​)=(mMy​​,mMx​​)⋅My​ is the moment about the y-axis, given by My​=∬D​xρ(x,y)dA. We, therefore, have My​=∫01​∫0ex​31xydydx=∫01​ Step 5 The integral ∫01​231​xe2xdx can be calculated by parts. Using u=x and dv=231​ ∫01​231​xe2xdx=[431​xe2x−]01​= dx we get Step 6 Similarly, Mx​ is the moment about the x-axis, given by Mx​=∬D​yρ(x,y)dA. We, therefore, have Mx​=∫01​∫0ex​31y2dydx=∫01​xdx=