# The eld is conservative 637 chapter 17 22 a let r d

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Unformatted text preview: θ dθ = 1/3 π/4 16 (3π − 4) 9 581 Chapter 16 π /2 1 0 0 2π 2 dθ = π/8 0 e−r r dr dθ = 2 24. 0 π /2 1 4 r3 dr dθ = 23. 0 π /2 2 cos θ 0 0 π /2 cos3 θ dθ = 16/9 0 1 0 0 π /2 a 0 π /4 sec θ tan θ r2 dr dθ = 28. 0 0 π /4 2 29. 0 dθ = 0 π sin 1 4 π r 1 − 1/ 1 + a2 dr dθ = 2 (1 + r2 )3/2 27. 0 π /2 1 sin 1 2 cos(r2 )r dr dθ = 26. dθ = (1 − e−4 )π 0 π /2 8 3 r2 dr dθ = 25. 2π 1 (1 − e−4 ) 2 √ 0 1 3 √ sec3 θ tan3 θ dθ = 2( 2 + 1)/45 0 π√ r dr dθ = ( 5 − 1) 4 1 + r2 π /2 5 r dr dθ = 30. tan−1 (3/4) 3 csc θ = 2π a 1 2 π /2 tan−1 (3/4) 2π 0 h 0 π /2 a 32. (a) V = 8 0 0 (25 − 9 csc2 θ)dθ 25 25 π − tan−1 (3/4) − 6 = tan−1 (4/3) − 6 22 2 hr dr dθ = 31. V = 0 π /4 a2 dθ = πa2 h 2 4c c2 (a − r2 )1/2 r dr dθ = − π (a2 − r2 )3/2 a 3a a = 0 42 πa c 3 4 (b) V ≈ π (6378.1370)2 6356.5231 ≈ 1,083,168,200,000 km3 3 π /2 a sin θ 33. V = 2 0 0 2 c2 (a − r2 )1/2 r dr dθ = a2 c a 3 √ a 2 cos 2θ π /4 0 cos 2θ dθ = 2a2 0 π /4 0 4 sin θ 35. A = √ 8 cos 2θ π/6 π /2 4 sin θ r dr dθ + r dr dθ π/4 π /4 0 π /2 (8 sin2 θ − 4 cos 2θ)dθ + = π/6 √ 8 sin2 θ dθ = 4π/3 + 2 3 − 2 π/4 φ 2a sin θ φ r dr dθ = 2a2 36. A = 0 (1 − cos3 θ)dθ = (3π − 4)a2 c/9 0 π /4 r dr dθ = 4a2 34. A = 4 π /2 0 0 1 sin2 θ dθ = a2 φ − a2 sin 2φ 2 Exercise Set 16.4 582 +∞ 37. (a) I 2 = +∞ e−x dx 2 0 +∞ e−y dy = 2 0 +∞ +∞ = 0 0 e−x e−y dx dy = 2 2 +∞ π /2 0 0 e−r r dr dθ = 2 0 1 2 +∞ e−(x 2π 2π R D(r)r dr dθ = 0 0 0 tan−1 (2) tan−1 (1/3) dx dy (c) I= 1 1 re−r dr dθ = π 4 0 re−r dr ≈ 1.173108605 4 0 ke−r r dr dθ = −2πk (1 + r)e−r R = 2πk [1 − (R + 1)e−R ] 0 tan−1 (2) 2 cos2 θ dθ = tan−1 (1/3) 0 √ π/2 0 r cos θ dr dθ = 4 40. +y 2 ) 0 2 3 2 dθ = π/4 (b) R 2 π /2 0 39. V = 2 0 π 38. (a) 1.173108605 e−x dx e−y dy 0 +∞ 0 (b) I 2 = +∞ 1 1π + 2[tan−1 (2) − tan−1 (1/3)] = + 5 5 2 EXERCISE SET 16.4 z 1. (a) z (b) y x x y z (c) x y z 2. (a) z (b) x y y x 583 Chapter 16 z (c) y x 3. (a) x = u, y = v, z = 53 + u − 2v 22 (b) x = u, y = v, z = u2 4. (a) x = u, y = v, z = v 1 + u2 (b) x = u, y = v, z = 12 5 v− 3 3 5. (a) x = 5 cos u, y = 5 sin u, z = v ; 0 ≤ u ≤ 2π, 0 ≤ v ≤ 1 (b) x = 2 cos u, y = v, z = 2 sin u; 0 ≤ u ≤ 2π, 1 ≤ v ≤ 3 6. (a) x = u, y = 1 − u, z = v ; −1 ≤ v ≤ 1 7. x = u, y = sin u cos v, z = sin u sin v 9. x = r cos θ, y = r sin θ, z = 1 1 + r2 (b) x = u, y = 5 + 2v, z = v ; 0 ≤ u ≤ 3 8. x = u, y = eu cos v, z = eu sin v 10. x = r cos θ, y = r sin θ, z = e−r 2 11. x = r cos θ, y = r sin θ, z = 2r2 cos θ sin θ 12. x = r cos θ, y = r sin θ, z = r2 (cos2 θ − sin2 θ) 13. x = r cos θ, y = r sin θ, z = √ √ 9 − r2 ; r ≤ 5 14. x = r cos θ, y = r sin θ, z = r; r ≤ 3 √ 1 1 3 ρ 15. x = ρ cos θ, y = ρ sin θ, z = 2 2 2 16. x = 3 cos θ, y = 3 sin θ, z = 3 cot φ 17. z = x − 2y ; a plane 18. y = x2 + z 2 , 0 ≤ y ≤ 4; part of a circular paraboloid 19. (x/3)2 + (y/2)2 = 1; 2 ≤ z ≤ 4; part of an elliptic cylinder 20. z = x2 + y 2 ; 0 ≤ z ≤ 4; part of a circular paraboloid 21. (x/3)2 + (y/4)2 = z 2 ; 0 ≤ z ≤ 1; part of an elliptic cone 22. x2 + (y/2)2 + (z/3)2 = 1; part of an ellipsoid 23. (a) x = r cos θ, y = r sin θ, z = r; x = u, y = v, z = √ u2 + v 2 ; 0 ≤ z ≤ 2 24. (a) I: x = r cos θ, y = r sin θ, z = r2 ; II: x = u, y = v, z = u2 + v 2 ; u2 + v 2 ≤ 2 25. (a) 0 ≤ u ≤ 3, 0 ≤ v ≤ π (b) 0 ≤ u ≤ 4, −π/2 ≤ v ≤ π/2 Exercise Set 16.4 584 26. (a) 0 ≤ u ≤ 6, −π ≤ v ≤ 0 (b) 0 ≤ u ≤ 5, π/2 ≤ v ≤ 3π/2 27. (a) 0 ≤ φ ≤ π/2, 0 ≤ θ ≤ 2π (b) 0 ≤ φ ≤ π, 0 ≤ θ ≤ π 28. (a) π/2 ≤ φ ≤ π (b) 0 ≤ ≤ π/2, 0 ≤ φ ≤ π/2 29. u = 1, v = 2, ru × rv = −2i − 4j + k; 2x + 4y − z = 5 30. u = 1, v = 2, ru × rv = −4i − 2j + 8k; 2x + y − 4z = −6 31. u = 0, v = 1, ru × rv = 6k; z = 0 32. ru × rv = 2i − j − 3k; 2x − y − 3z = −4 √ √ 33. ru × rv = ( 2/2)i − ( 2/2)j + (1/2)k; x − y + √ √ π2 2 z= 2 8 34. ru × rv = 2i − ln 2k; 2x − (ln 2)z = 0 35. z = 2 2 9 − y 2 , zx = 0, zy = −y/ 9 − y 2 , zx + zy + 1 = 9/(9 − y 2 ), 2 3 S= −3 0 2 3 9 − y2 dy dx = 3π dx = 6π 0 4−x 4 2 2 36. z = 8 − 2x − 2y , zx + zy + 1 = 4 + 4 + 1 = 9, S = 4 3(4 − x)dx = 24 3 dy dx = 0 0 0 37. z 2 = 4x2 + 4y 2 , 2zzx = 8x so zx = 4x/z , similarly zy = 4y/z thus x 1 2 2 zx + zy + 1 = (16x2 + 16y 2 )/z 2 + 1 = 5, S = √ 5 dy dx = √ 5 x2 0 1 (x − x2 )dx = 0 2 2 38. z 2 = x2 + y 2 , zx = x/z , zy = y/z , zx + zy + 1 = (z 2 + y 2 )/z 2 + 1 = 2, π /2 2 cos θ √ π /2 √ √ √ S= 2 dA = 2 2 r dr dθ = 4 2 cos2 θ dθ = 2π 0 R 0 0 2 2 39. zx = −2x, zy = −2y , zx + zy + 1 = 4x2 + 4y 2 + 1, 2π 1 4x2 + 4y 2 + 1 dA = S= r 0 R = 4r2 + 1 dr dθ 0 1√ (5 5 − 1) 12 2π √ dθ = (5 5 − 1)π/6 0 2 2 40. zx = 2, zy = 2y , zx + zy + 1 = 5 + 4y 2 , y 1 1 5 + 4y 2 dx dy = S= 0 0 y √ 5 + 4y...
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## This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

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