Find the center of mass of the lamina that has the given shape and density. y = sqrt3(x) x, y = 0 , x = 4; ρ(r, θ) = r2

Use cylindrical coordinates. Evaluate x2 + v2 dv, where E is the region that lies inside the cylinder x2 + y²2 = 1 and between the planes z = -6 and z = -1.

Geographers measure the geographical center of a country (which is the centroid) and the population center of the country (which is the center of mass computed with the population density). A hypothetical country is shown in the figure to the right with the location and population of five towns. Assuming no one lives outside the towns, find the geographical center of the country and the population center of the country. ▬▬ My = S S dy dx + S S dy dx + [10**- || | ▬▬ SSO dy dx 00 (Type exact answers.) (-16,16) (-8,8) Pop.= 10,000 (-11,-8) Pop. 15.000 F (-8,-8) (8-8) (16,16) (8.15) Pop. 15,000 (-16,-16) For the geographical center, determine the double integrals to be used to most efficiently find M,, the region's first moment about the y-axis. For the geographical center calculations, assume a density of 1. Use increasing limits of integration. Divide the region into three sections, going from left to right. (8,0) Pop=20,000 (16,-16) Pop.= 4,000

Use cylindrical coordinates. Find the mass and center of mass of the S solid bounded by the paraboloid z = 10x2 + 10y2 and the plane z = a (a > 0) if S has constant density K. m = (7. v. 3) = (|