Evaluate (x + y) dA, given that R is the region inside a triangle with vertices...

   

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Evaluate (x² + y) dA, given that R is the region inside a triangle with vertices (0.0), (1.1), and (1,-1). A lamina is a sector of a circle radius r with an angle ß as shown in Figure 1. Figure 1 If the lamina has constant density, use double integral in polar coordinates to answer the followings: (i) Show that the area of the lamina is r² B. (B in radians). (ii) Find the centroid of the lamina. (Transform the integral √12 MM √12-x² √2 √16-x²-y2 into each of the following coordinates systems: (i) cylindrical, dz dy dx (ii) spherical. Hence, evaluate the integral using only one of the coordinates system. 33. (a) 1 12 1³h 1.² L f (c) (i) 16-72 2a (b) (ii) x = sin ß. y = (1 - cos ß) 38 r dz dr d0. (ii) 2a 3B 2л 1²* f* fp² sino dp de = 2sec 40л 3 Evaluate (x² + y) dA, given that R is the region inside a triangle with vertices (0.0), (1.1), and (1,-1). A lamina is a sector of a circle radius r with an angle ß as shown in Figure 1. Figure 1 If the lamina has constant density, use double integral in polar coordinates to answer the followings: (i) Show that the area of the lamina is r² B. (B in radians). (ii) Find the centroid of the lamina. (Transform the integral √12 MM √12-x² √2 √16-x²-y2 into each of the following coordinates systems: (i) cylindrical, dz dy dx (ii) spherical. Hence, evaluate the integral using only one of the coordinates system. 33. (a) 1 12 1³h 1.² L f (c) (i) 16-72 2a (b) (ii) x = sin ß. y = (1 - cos ß) 38 r dz dr d0. (ii) 2a 3B 2л 1²* f* fp² sino dp de = 2sec 40л 3

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i Show that the area of the lamina is r in radians To show that the area of the lamina is r we can use the following double integral in polar coordina... View the full answer