# Z 2 1 1 2 3 x 1 1 y 1 1 z 1 b evaluate

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z 2 = 1. 0 1 2 3 x -1 0 1 y -1 0 1 z 1 (b) Evaluate integraldisplay γ F · d s where F ( x, y ) = ( e x 2 + y, x 2 ) and γ is a counterclockwise parametrization of the boundary of the first quadrant region below the graph of y = 1 x 3 . (c) Evaluate integraldisplay γ ω , where ω = y x 2 + y 2 dx x x 2 + y 2 dy and γ is a clockwise parametriza- tion of the circle x 2 + y 2 2( y + 1) = 0. (d) Evaluate integraldisplay γ F · d s , where F ( x, y, z ) = ( xy, 0 , z ) and γ is the straight line segment joining (0 , 1 , 0) to (1 , 2 , 3).

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MATB42H page 2 6. [10 points] The parametrized surface, Φ ( u, v ) = (sin u, sin(2 u ) , v ), 0 u 2 π , 1 v 1, is shown on the right. (a) Draw the u –line, γ ( u ) = Φ ( u, 0) on the diagram. 0 1 x 0 1 y -1 0 1 z (b) Find the area of the region of the xy –plane bounded by γ ( u ) (from part (a)). (c) Find an equation for the tangent plane to Φ at Φ parenleftbigg 3 π 4 , 1 2 parenrightbigg = parenleftbigg 1 2 , 1 , 1 2 parenrightbigg . 7. [10 points] The ellipse 4( z 1) 2 + y 2 = 1 in the yz –plane is rotated about the y –axis to produce a surface S in R 3 . (a) Parametrize S . (b) Set up, but do not evaluate, the integral which would give the surface area of S . (c) Let ω = x r 3 dy dz + y r 3 dz dx + z r 3 dx dy , where r = radicalbig x 2 + y 2 + z 2 . If S is oriented by the outward pointing unit normal, evaluate integraldisplay S ω . 8. [24 points] (a) Let S be the piece of the graph of z = 4 x 2 y 2 which lies over the region of the xy –plane bounded by the polar graph r = 1 + cos θ , oriented by the upward pointing unit normal. Evaluate integraldisplay
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