北大版高数第八章习题解答.pdfVIP

5p Æ6 1lÙ SK ‰ SSSKKK8.1 1.  L (xy+yz+zx)ds, ¥L ǑL o :O(0, 0, 0), A(0, 0, 1), B (0, 1, 1), C (1, 1, 1) ò ‚. . I = 1 0dz + 1 ydy + 1(x + 1 + x)dx = 5 . 0 0 0 2 xyds, 2.  L ¥L  / |x | + |y | = a (a 0). .  È ¼ ux ¼ , È© ‚L uy¶ é¡ , dÈ© Ǒ0. 2 3.  L (1+y )ds, ¥L Ǒ{ ‚ã : x = a(t − sin t), y = a(1 − cos t), 0 ≤ t ≤ 2π . 2π 2 4 t t 256 3 . I = 0 (1 + 4a sin 2 )2a sin 2 dt = 8a + 15 a . 1 4.  L x2+y2+z2 ds, ¥L ǑÚ ^ ‚ã : x = a cos t, y = a sin t, z = bt, 0 ≤ t ≤ 2π . 2π 1 √ 2 2 √a2+b2 2πb . I = 2 2 2 a + b dt = arctan . 0 a +b t ab a 2 2 5.  C (x + y)ds, ¥C ǑV ‚r = a cos2θ m ¡‹ . . ‚C ë § : x = a√cos2θ cos θ, y = a√cos2θ sin θ, −π ≤ θ ≤ π . 4 4 ′2 ′2 a2 π √ a √ 2 4 x + y = cos2θ . uI = − π a cos2θ(cosθ + sin θ) √cos2θ dθ = 2a . 4 2 2 6.  xyds, ¥Lý ± x2 + y2 = 1 u 1 – ¥Ü © . L a b . Lë § : x = a cos θ, y = b sin θ, 0 ≤ θ ≤ π . 2 π 2 ab(a3 −b3) 2 2 2 2 I = 0 ab cos θ sin θ a sin θ + b cos θdθ = a2−b2 . 2 2 7.  L x + y ds, ¥L Ǒ ‚ãx = a(cost + t sin t), y = a(sint − t cos t), (0 ≤ t ≤ 2π). 3