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# Tutorial_8 - = x 2 y y 2 z K 5 Evaluate Z C g x,y,z ds...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 8 1. Find the area of the surface consisting of the part of the sphere of radius 2 centered at origin that lies above the horizontal plane z = 1. (Equation of this sphere is given by x 2 + y 2 + z 2 = 2 2 .) Ans : 4 π 2. Find the centre of mass of the lamina of density ρ ( x, y ) = x 2 that occupies the region R bounded by the parabola y = 2 - x 2 and the line y = x . Ans : ( - 8 / 7 , - 20 / 49) 3. Evaluate the following triple integral: ZZZ D ( x 2 + 2 z ) dV , D is the solid cube {- 1 2 x 1 2 , - 1 2 y 1 2 , - 1 2 z 1 2 } . Ans : 1 12 4. Let F ( x, y, z ) = 2 xy i + ( x 2 + 2 yz ) j + y 2 k . Show that F is a conservative vector field. Find a function f such that f = F . Ans : f ( x, y, z ) = x 2
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Unformatted text preview: ) = x 2 y + y 2 z + K 5. Evaluate Z C g ( x,y,z ) ds , where g ( x,y,z ) = x 2-yz + z 2 and C is the line segment from (0 , , 0) to (1 , 2 , 3). Ans : 4 √ 14 / 3 6. Compute the work done by the force F ( x,y,z ) = yz i + 2 y j-x 2 k on a particle that moves along the curve C given by the vector function r ( t ) = t i + t 2 j + t 3 k , for 0 ≤ t ≤ 1. Ans : 17 / 30 7. Evaluate Z C 2 xy dx + ( x 2 + z ) dy + y dz , where C consists of two line segments: C 1 from (0 , , 0) to (1 , , 2), and C 2 from (1 , , 2) to (3 , 4 , 1). Ans : 40...
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