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課程名稱︰微積分甲下
課程性質︰必修
課程教師︰康明昌
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2011.06.17
考試時限（分鐘）：150分左右
是否需發放獎勵金:是
（如未明確表示，則不予發放）
試題 :
滿分130
You may use all the properties of gamma function (Γ(x)) and beta function
(Β(x,y)) (including Γ(1/2)=√π) without proving them.
(1) Find the volume of the solid bounded by the surface z=2-x^2-y^2 and the
xy-plane. (15%)
(2)Evaluate the integral ∫ 1/(1+x^2+y^2) dA
D
Where D is one loop of the lemniscate (x^2+y^2)^2-(x^2-y^2)=0 (15%)
∞ ∞
(3)Evaluate ∫ exp(-x^2) dx and ∫ (x^2)*exp(-(x-μ)^2/2σ^2) dx
0 -∞
Where σ>0, μ and σ are constants. (20%)
4
(4)Find the volume of the simplex E in R which is bounded by the hyperplane
x1/a1+x2/a2+x3/a3+x4/a4=1 and the planes defined by xi=0 for 1≦i≦4 where
a1,a2,a3,a4>0 (20%)
3
(5)Let S be the spherical cap of the part of the sphere {(x,y,z)belongs to R :
x^2+y^2+z^2=1} with 1/2≦x≦1
→ → →
Let F=-xi +zk be avector field and n be the outward normal.
→
Find ∫ F‧ｎ dσ (20%)
S
3 → → →
(6)Let E be the cube in R defined by 1≦x,y,z≦2, F=xzi+xyzj-y^2k be a vector
field. Let S1 and S2 be the faces of E defined by x=1 and z=2 respectively.
→ → →
Find ∫ curl F‧ｎ dσ +∫ curl F‧ｎ dσ where n is the outward normal.(20%)
S1 S2
2 3
(7) Let S ={(x,y,z)belong to R :x^2+y^2+z^2=1}
Suppose ∫ (x^6-2y^6+5z^6)dσ=a/7*π
S2
Find the value a. (20%)
π
(Hint: It is NOT difficult to evaluate ∫(sinθ)^5 dθ.
0
Of cource, you may evaluate the integral ∫ f(x,y,z)dxdydz by other
D3
easier(!) methods.)
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※ 編輯: allen2759393 來自: 140.112.196.120 (06/20 17:52)
※ 編輯: allen2759393 來自: 140.112.196.120 (06/20 17:54)