notes001 (29)

notes001 (29) - part 1. Solution: The exact value of the...

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MATH 2203 - Quiz 5 (Version 2) Solution November 2, 2009 NAME______________________ 1. (a) Use a Riemann sum with m = n = 2 to estimate the value of ZZ R cos ( x + y ) dA where R = [0 ] [0 ] . Take the sample points to be the upper right corners. Solution: We have x = ± 0 2 = 2 and likewise y = 2 . Therefore A x y = 2 4 . The upper right corners of the subrectangles are ( 2 2) ; ( 2) ; ( 2 ) ; and ( ) . Therefore the Riemann sum estimate is 2 4 cos 2 + 2 ± + cos + 2 ± + cos 2 + ± + cos ( + ) ± = 0 . (b) Obtain another estimate for this integral by using the midpoints as sample points. Solution: The midpoints of the subrectangles are ( 4 4) ; (3 4 4) ; ( 4 ; 3 4) ; and (3 4 ; 3 4) . Therefore the Rie- mann sum estimate is 2 4 ² cos 4 + 4 ± + cos ² 3 4 + 4 ³ + cos ² 4 + 3 4 ³ + cos ² 3 4 + 3 4 ³³ = 2 4 ( ± 2) = ± 2 2 . 1
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Unformatted text preview: part 1. Solution: The exact value of the given integral is Z & Z & cos ( x + y ) dy dx . Evaluating the inner integral, we obtain Z & cos ( x + y ) dy = sin ( x + y ) j y = & y =0 = sin ( x + & ) & sin ( x ) = sin ( x ) cos ( & ) + cos ( x ) sin ( & ) & sin ( x ) = & 2 sin ( x ) . This gives Z & Z & cos ( x + y ) dy dx = & Z & 2 sin ( x ) dx = 2 cos ( x ) j x = & x =0 = 2 cos ( & ) & 2 cos (0) = & 4 . Here is a picture of f ( x; y ) = cos ( x + y ) . Note that most of the graph lies below the plane z = 0 , which explains why the integral is a negative number. 2 3...
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notes001 (29) - part 1. Solution: The exact value of the...

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