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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online