91ç-ç‚Bø ‚5

91ç-ç‚Bø ‚5

1. q a, b, c îÑ£b, D ÑÞ x a + y

b + z

c = 1 Dúè™ÞFˇA5ûÞñ, t° D 5$-(centroid) è™
(12%)

2. q DÑÀP7 x²+ y²+ z² ≤ 1
(i) t°ú½ }

ZZZ

D

|z|³ dV 5M
(8%)

(ii) t°ú½ } ZZZ

D

|x + y + z|³ dV 5M
(6%)

3. q R ÑJ (1, 0), (5, −4) £ (5, 0) ÑÝõíúi$–, t°ù½ } ZZ

R

x− y x+ y dxdy 5M
(12%)

4. q C Ñ x² + z² = 1 D y = x² s6Þ5>(, t°( } Z

C

√1 + 4x²z² ds 5 M
(12%)

5. q −→F(x, y) = − y

x²+ y²− 2x + 1

−

→i + x− 1 x²+ y²− 2x + 1

−

→j , C¹ [( −→γ1(t) =

2 cos³t−→i + sin t−→j , C² [( −→γ2(t) = 1

2cos³t−→i + sin t−→j , 0 ≤ t ≤ 2π, t°

(i) −→F 5ì −→

∇ ×−→F
(4%) (ii)

Z

C₁

−

→F ·−→T ds D Z

C₂

−

→F ·−→T ds 5M
(10%)

6. q S [Þ y = x², 0 ≤ x ≤ 1, 0 ≤ z ≤ 3, t°Þ } ZZ

S

x dσ 5M
(12%)

7. qZZ−→F(x, y, z) = 3y−→i − xz−→j + yz²−→k, S [Þ 2z = x² + y², z ≤ 2, t°Þ }

S

−

→∇ ×−→F · −→n dσ 5M, w2 −→n ²-
(12%)

8. q −→F(x, y, z) = x³−→i + 3yz²−→j + (3y²z+ x²)−→k, S Ñ7Þ x²+ y²+ z² = a², t°

Þ } ZZ

S

−

→F · −→n dσ 5M, w2 −→n ²Õ
(12%)

1