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Tutorial 8

# Tutorial 8 - x t = t 2 t 3 t 0 ≤ t ≤ 2 6 Find the work...

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MATH1211/10–11(1)/Tu8 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010–11 First Semester: Tutorial 8 1. Rewrite Z 1 0 Z x 0 sin x dy dx + Z 2 1 Z 2 - x 0 sin x dy dx into a single iterated integral by reversing the order of integration, and evaluate. 2. Change the order of integration of Z 1 0 Z 2 0 Z x 2 0 f ( x, y, z ) dz dx dy to give five other equivalent iterated integrals. 3. Evaluate RR D (2 x + y ) 2 e x - y dA , where D is the region enclosed by 2 x + y = 1 , 2 x + y = 4 , x - y = - 1, and x - y = 1. 4. Find the average value of g ( x, y, z ) = e z over the unit ball given by B = { ( x, y, z ) | x 2 + y 2 + z 2 1 } . 5. Calculate R x f ds where f ( x, y, z ) =
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Unformatted text preview: , x ( t ) = ( t, 2 t, 3 t ), 0 ≤ t ≤ 2. 6. Find the work done by the force ﬁeld F = x 2 y i + z j +(2 x-y ) k on a particle as the particle moves along a straight line from (1 , 1 , 1) to (2 ,-3 , 3). 7. Verify Green’s theorem for the given vector ﬁeld F = M ( x,y ) i + N ( x,y ) j and the region D by calculating both I ∂D M dx + N dy and ZZ D ( N x-M y ) dA : F = ( x 2-y ) i + ( x + y 2 ) j , D is the rectangle bounded by x = 0 , x = 2 , y = 0, and y = 1. 1...
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