# Is the line segment from 0 1 to 1 1 20 compute r c x

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is the line segment from (0 , - 1) to (1 , 1) . 20. Compute R C ( x + 2 y + z ) ds, where C is the helix ~ r ( t ) = (cos t, sin t, t ) for 0 t 3 . 21. Compute R C ~ F · d~ r, where ~ F ( x, y ) = h 2 xy, x 2 + y 2 i and C is the part of the unit circle in the first quadrant, oriented counterclockwise. 22. Compute R C ~ F · d~ r, where ~ F ( x, y, z ) = h x - y, y - z, z i and C is the line segment from (0 , 0 , 0) to (1 , 1 , 4) . 23. Determine if ~ F is conservative, and if so, find a potential function f . (a) ~ F ( x, y ) = h 4 x 3 y 5 , 5 x 4 y 4 i (b) ~ F ( x, y, z ) = h xyz, 1 2 x 2 z, 2 z 2 y i (c) ~ F ( x, y, z ) = h cos z, 2 y, - x sin z i 24. Let ~ F = f, for f ( x, y, z ) = xye z (so ~ F is conservative!). Compute R C ~ F · d~ r , where: (a) C is any curve from (1 , 1 , 0) to (3 , e, - 1) . (b) C is the boundary of the unit square, oriented counterclockwise. 25. Use Green’s Theorem to compute H C xy 3 dx + x 3 y dy, where C is the rectangle [ - 1 , 2] × [ - 2 , 3] , oriented counterclockwise. 26. Use Green’s Theorem to compute H C y 2 dx - x 2 dy, where C consists of the arcs y = x 2 and y = x , 0 x 1 , oriented clockwise. [Since it is clockwise, recall that R - C ~ F · d~ r = - R C ~ F · d~ r ] Answers Page 2 of 4
MATH 223 Review for Final Part I Spring 2016 1. (a) Max: f (0 , 2) = f (2 , 0) = 8 ; min: f (1 , 1) = - 1 (b) Max: f (2 , 4) = 10 ; min: f ( - 2 , 4) = - 18 2. (a) Max: f 6 13 , - 4 13 = 26 13 ; min: f - 6 13 , 4 13 = - 26 13 (b) There are six critical points: (0 , 2) , (0 , - 2) , 6 , 2 3 , 6 , - 2 3 , - 6 , 2 3 , - 6 , - 2 3 . Max: f 6 , 2 3 = f - 6 , 2 3 = 12 3 ; min: f 6 , - 2 3 = f - 6 , - 2 3 = - 12 3 3. 32 / 3 4. 1 6 (1 - e ) 5. (a) 3 e 12 + e 8 (b) (sin 1) ln 2 6. (a) R 1 0 R y y 2 f ( x, y ) dx dy (b) R 3 0 R 9 x 2 f ( x, y ) dy dx 7. - 1 2 + π 4 8. 9 / 2 9. 4 / 3 ; One way to set up the integral is Z 2 0 Z 1 - x 2 0 Z 2 - x - 2 y 0 4 x dz dy dx 10. 3 ; If you project onto the yz -plane, your integral should be R 1 0 R 3 2 - z 2 z R 4 0