C3PracticeTest3Sol

# C3PracticeTest3Sol - MA 242 Test 3 Solutions 1[10 points If...

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MA 242 Test 3 Solutions 1. [10 points] If f is a constant function, f ( x,y ) = k , and R = [ a,b ] × [ c,d ], compute ZZ R kdA . Which theorem was used in this problem? Solution: Z d c Z b a kdxdy = k Z d c dy Z b a dx = k · [ y ] d c · [ x ] b a = k ( d - c )( b - a ) We had to use Fubini’s Theorem to compute this integral. 2. [15 points] Evaluate the integral, Z 1 0 Z 1 y p x 3 + 1 dxdy , by reversing the order of integration. Solution: If we look at the integral in its given Type II form we see that D = { ( x,y ) | 0 y 1 , y x 1 } . This region can also be described in Type I form. In Type I form, D = { ( x,y ) | 0 x 1 , 0 y x 2 } . Then Z 1 0 Z 1 y p x 3 + 1 dxdy = Z 1 0 Z x 2 0 p x 3 + 1 dydx First compute the inner integral, R x 2 0 x 3 + 1 dy = [ y x 3 + 1] y = x 2 y =0 = x 2 x 3 + 1 Next compute the outer integral via u-substitution,(choose u = x 3 + 1, then du = 3 x 2 dx ) Z 1 0 x 2 p x 3 + 1 dx = [ 2 9 ( x 3 + 1) 3 / 2 ] x =1 x =0 = 2

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C3PracticeTest3Sol - MA 242 Test 3 Solutions 1[10 points If...

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