vcprac03s - The University of Sydney School of Mathematics...

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics Solutions to Practice Session 3 MATH2061: Vector Calculus Summer School 2012 1. Sketch the following regions: (a) { ( x, y ) ∈ R 2 | 1 ≤ x ≤ 2 , − x ≤ y ≤ x } (b) { ( x, y ) ∈ R 2 | x = r cos θ, y = r sin θ, ≤ θ ≤ 2 π, 1 ≤ r ≤ 2 } Solution: (a) 1 2 (b) 1 2 2. Describe the shaded regions as sets of points in R 2 (as in question 1 ). Use polar coordinates in part (c). (a) (b) (c) y =2 x 3 y = x 2 2 y =1- x y = x 2 Solution: (a) { ( x, y ) ∈ R 2 | ≤ x ≤ 3 , ≤ y ≤ 2 x } (b) The points of intersection of the parabola and the line are ( 1 2 , 1 4 ) and ( − 1 , 1). The shaded region is { ( x, y ) ∈ R 2 | − 1 ≤ x ≤ 1 2 , x 2 ≤ y ≤ 1- x 2 } . (c) { ( x, y ) ∈ R 2 | x = r cos θ, y = r sin θ, ≤ θ ≤ π/ 4 , ≤ r ≤ 2 } 3. Sketch the region of integration and then evaluate integraldisplay 2 integraldisplay 3 ( x − 2) sin y dx dy ....
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vcprac03s - The University of Sydney School of Mathematics...

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