s07wk10 - Math 23 B Dodson Week 10 Homework 15.5...

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Math 23 B. Dodson Week 10 Homework: 15.5 Applications (mass, center of mass) 15.6 Surface Area Problem 15.5.6: Find the mass and center of mass of a thin plate (lamina) occupying the triangular region with verticies at (0 , 0) , (1 , 1) and (4 , 0) , if the density at ( x, y ) is ρ ( x, y ) = x. Solution: We have mass m = ZZ D ρ ( x, y ) dA, and need an iterated integral to evaluate the double integral. Checking the sketch, we see that the region is of Type II, with 0 y 1 , and y x 4 - 3 y (where the line from (1 , 1) to (4 , 0) has slope - 1 3 , and we solve y - 0 = - 1 3 ( x - 4) for x ). The iterated integral is then Z 1 0 Z 4 - 3 y y x dxdy. Evaluation gives m = 10 3 . For the center of mass, ( x, y ) , we have x = 1 m ZZ D ( x, y ) dA,
2 and y = 1 m ZZ D ( x, y ) dA. The iterated integrals have the same limits as above, but in

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