課程名稱︰微積分甲下
課程性質︰大一必修
課程教師︰容志輝
開課學院：工學院
開課系所︰電機系、材料系
考試日期（年月日）︰2015/06/15
考試時限（分鐘）：50 分鐘
試題 :
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103-2 Calculus class-2 4th test
2 2
1. Consider the vector field: F(x,y) = (3 + 2xy)i + (x - 3y )j
(a) Show that F is conservative by finding a potential function for it.
(b) Evaluate the line integral ∫ F．dr where C is the curve given by
C
t t
r(t) = e sin(t) i + e cos(t) j 0 <= t <= π
x
2. Evaluate ∮ e [ (1-cos(y))dx - (y-sin(y))dy ], C is the boundary of
C
0 <= x <= π, 0 <= y <= sin(x) oriented counterclockwise.
2 2
3. Consider F(x,y,z) = -y i + x j + z k
(a) Find curl F
(b) Evaluate ∫ F．dr, where C is the curve of intersection of the plane
C
2 2
y + z = 2 and cylinder x + y = 1.
4.
1 √3
(a) consider r(t) = (─(1 + cos(t)) , ──(1 + cos(t)) , sin(t) ), where
2 2
π
0 <= t <= ─, find the centroid of r(t).
2
(b) consider r(t,u) = ( (1+cos(t))sin(u) , (1+cos(t))cos(u) , sin(t) ), where
π
0 <= t <= ─, 0 <= u <= 2π, find the surface area of r.
2
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