[試題] 107-2 余正道 微積分二 期末考
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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