Final answer:
To find the mass and center of mass of a lamina with a varying density function within region D, set up double integrals to compute the total mass and the first moments, then divide the moments by the total mass.
Step-by-step explanation:
To find the mass and center of mass of a lamina with a given density function, one needs to use calculus to integrate the density function over the given region. For a lamina that occupies the region D bounded by y = x1/2 and y = x2, with a density function σ(x, y) = kx2, you would set up double integrals over the region D to compute the total mass and the moments about the x and y axes. The mass (M) can be found by integrating the density function over D. The center of mass (xcm, ycm) can then be found by dividing the first moments by the total mass, specifically xcm = Mx/M and ycm = My/M, where Mx and My are the first moments of the mass distribution about the y-axis and the x-axis, respectively.