vctut03 - The University of Sydney School of Mathematics...

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The University of Sydney School of Mathematics and Statistics Tutorial 3 MATH2061: Vector Calculus Summer School 2012 1. Let I = integraldisplayintegraldisplay R xydxdy where R is the region in the xy plane bounded by y = x 2 , x = 0, y = 4 and where x 0. (a) Sketch the region R . (b) Evaluate I by integrating with respect to y first. (c) Evaluate I by integrating with respect to x first. 2. Sketch the region of integration and evaluate integraldisplay 1 0 integraldisplay 2 x x e x + y dydx. 3. Describe the region R , and use polar co-ordinates to evaluate the following: integraldisplayintegraldisplay R ( x 2 + y 2 ) 1 / 2 dA, R : x 2 + y 2 4 . 4. The sphere x 2 + y 2 + z 2 = 25 has a hole bored through it by the cylinder x 2 + y 2 = 4. Find the volume of the sphere that is removed. 5. Evaluate contintegraldisplay C ( x 2 xy 3 ) dx + ( y 2 2 xy ) dy, where C is the square with vertices at (0 , 0) , (2 , 0) , (2 , 2) , (0 , 2) and the contour is traversed in the anticlockwise direc- tion. 6. Evaluate contintegraldisplay C F · d r where F = (tan - 1 x + y 3 ) i (3 x 2 + sin y ) j , and C is the curve made up of the straight line segments from (0
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