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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