課程名稱︰微積分二 課程性質︰數學系大一必修 課程教師︰余正道老師 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2019/6/13 考試時限（分鐘）：180 試題 : There are 119 points in total 1. [10%] Suppose S, T has area and S is contained in (包含於) T. (a) Show that the areas satisfy A(S)≦A(T). (b) Show that T\S has area. 2. [10%] Evaluate the integral ∞ ∫ e^(-x^2)dx. -∞ 3. [20%] Let Σ be the surface cut out of the cylinder x^2+z^2=1 by x^2+y^2=1 and z≧0. (a) Find the surface area of Σ. (b) Find the volume of the region above the z-plane and below Σ. 4. [10%] Let Σ be the ellipsoid x^2 y^2 z^2 -----+-----+-----=1 with element of area dS. a^2 b^2 c^2 Let h: Σ→R denote the distance from the origin to the tangent plane of Σ at (x,y,z). Compute the surface integral ∫∫ h dS. Σ 5. [15%] Prove Gauss's theorem ∫boundary of R fdx+gdy=∫∫R (-fy+gx)dxdy directly in the following case that R is the region in R^2 bounded by a≦x≦b andα(x)≦y≦β(x) where α(x)≦β(x) are C^1 functions and f, g are C^1 functions in a domain containing R. 6. [20%] (a) Assuming the conditions for the divergence theorem hold for the region R given by a≦θ≦b, f(θ)≦r≦g(θ) in terms of polar coordinates, show that 1 ---∫boundary of R r^2 dθ equals the area of R. 2 (b) Compute the area of the region bounded by the lemniscate r^2=cos2θ. 7. [10%] Find the simple closed curve C contained in R^2, oriented counterclockwise, that maximizes the line integral ∫ y^3 dx+(3x-x^3) dy C 8. [24%] For t,x>0, let f(t,x)=1/x exp(-t/2(x+1/x)). (a) Show that for any integer n≧0 and real number a>0, the improper integral ∞ ∫ f(t,x)對t偏微分n次 dx converges uniformly on t≧a. 0 ∞ (b) Let K(t)=∫ f(t,x) dx. Show that it satisfies t^2K''+tK'-t^2K=0. 0 ∞ (c) Evaluate ∫ K(t) dt. 0 K(t) (d) Show that lim -------=-1. t→0 logt --

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