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s11wk10su - Math 23 B Dodson Week 5a Homework 15.5...

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Math 23 B. Dodson Week 5a Homework: 15.5 Applications of Double Integrals 15.6 Triple integrals 15.7 Cylindrical Coordinates; 15.8 Spherical Coordinates 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 = integraldisplayintegraldisplay 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 integraldisplay 1 0 integraldisplay 4 - 3 y y x dxdy. Evaluation gives m = 10 3 . For the center of mass, ( x, y ) , we have x = 1 m integraldisplayintegraldisplay D ( x, y ) dA,
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2 and y = 1 m integraldisplayintegraldisplay
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