The boundary of a lamina consists of the semicircles y= 1-x² and y = 25-x² together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is inversely proportional to its distance from the origin. Hint: use polar coordinates

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

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find the integrate(v . dr) of the following case: