# C what does it mean for the 1form ω f 1 dx 1 f 2 dx

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(c) What does it mean for the 1–form ω = F 1 dx 1 + F 2 dx 2 + · · · + F n dx n to be exact ?

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MATB42H3 page 2 5. [6 points] Find the arc length of the curve γ ( t ) = t, 1 2 t 2 , 1 3 t 3 between the origin, (0 , 0 , 0), and the point ( 2 , 2 2 , 8 3 ) . 6. [6 points] You are about to paint a picket fence. The base of the fence is in the xy –plane as the piece of the circle x 2 + y 2 = 25 which lies in the first quadrant of the xy –plane and the height of the fence at the point ( x, y ) is given by f ( x, y ) = 10 - x - y (meters). If only one side of the fence is to be painted, use a path integral to find the area of the painted surface. 7. [18 points] Evaluate the following line integrals. (a) Z γ x x 2 + y 2 dy - y x 2 + y 2 dx , where γ is the circle ( x - 1) 2 + ( y - 2) 2 = 4 oriented in the clockwise direction. (b) Z γ dx + z dy + x dz , where γ ( t ) = (sin t, t, cos t ), 0 t 2 π . (c) Z γ F · d s , where F ( x, y, z ) = ( yz, xz, xy ) and γ ( t ) = ( t, sin t, sin 2 t ) , 0 t π 2 . 8. [7 points] (a) Give a parametrization of the curve of intersection of the cone z = p x 2 + y 2 and the sphere x 2 + y 2 + z 2 = 2 x .
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