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

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics Tutorial 3 MATH2061: Vector Calculus 1. Let I = R Summer School 2012 xy dxdy where R is the region in the xy plane bounded by y = x2 , 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. 1 2x 2. Sketch the region of integration and evaluate 0 x ex+y dy dx. 3. Describe the region R, and use polar co-ordinates to evaluate the following: (x2 + y 2 )1/2 dA, R R : x2 + y 2 4. 4. The sphere x2 +y 2 +z 2 = 25 has a hole bored through it by the cylinder x2 +y 2 = 4. Find the volume of the sphere that is removed. 5. Evaluate C (x2 - xy 3 ) dx + (y 2 - 2xy) 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 direction. 6. Evaluate C F d r where F = (tan-1 x + y 3 ) i - (3x2 + sin y) j, and C is the curve made up of the straight line segments from (0, 0) to (2, 0) to (2, 1) to (1, 1) and back to (0, 0), taken once in an anticlockwise direction. Copyright c 2012 The University of Sydney 1 Extra questions for outside the tutorial 7. Describe the region R, and use polar co-ordinates to evaluate the following: xy 2 dA, R R : 0 x 2, 0 y (4 - x2 )1/2 . 8. Evaluate y 2 ex R 2 +y 2 dxdy where R is the region defined by x2 + y 2 9, y 0, x 0. 9. Find the volume under the surface z = x + y 2 and above the set R, where R is the rectangle in the xy-plane with corners (1, 1), (1, 3), (2, 3), (2, 1). 10. Find the volume under the surface z = x + y + 2 and above the set R, where R is the triangle in the xy-plane with corners (0, 0), (2, 0), (0, 4). 2 ...
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This note was uploaded on 02/06/2012 for the course MATH 2061 taught by Professor Notsure during the Three '09 term at University of Sydney.

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

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