課程名稱︰微積分乙下
課程性質︰必帶
課程教師︰張樹城
開課學院：生科院
開課系所︰生科系 生技系
考試日期（年月日）︰2016/06/23
考試時限（分鐘）：兩小時
試題 :
1.(20%)
(a)Find the volume of the region in the first octant bounded by the coordinate planes and the plane
x+y/2+z/3=1
(b)Find the volume of the region enclosed by the surfaces
z=x^2+3y^2 and z=8-x^2-y^2
2.(15%)
(a)By using the triple integral to find the volume of the solid sphere of radius 1
(b)By using (a) to find the volume of the solid ellipsoid region bounded by
x^2/a^2+y^2/b^2+z^2/c^2≦1
3.(20%)
(a)Evaluate
1 1/2
∫∫ e^x^2 dxdy
0 y/2
(b)Evaluate
∫∫ln√(x^2+y^2) dxdy
I
Here I={0＜m≦√(x^2+y^2)≦1} for a positive constant m
(c)Evaluate
∫∫ln√(x^2+y^2) dxdy
D
Here D={(x,y):0≦√(x^2+y^2)≦1} is the unit disc in the plane
4.(10%)
Evaluate
(2,3,-1)
∫ ydx+xdy+4dz
(1,1,1)
5.(15%)
(a)Find the line integral
∮-ydx+xdy over the ellipse r:x^2/a^2+y^2/b^2=1
r
(b)Find the area of the region enclosed by the ellipse x^2/a^2+y^2/b^2=1
6.(20%)
(a)Evaluate
∮y/(x^2+y^2)dx-x/(x^2+y^2)dy
Cr
Here Cr(t)=(rcost,rsint),0≦t≦2pi,is a circle of radius r
(b)Evaluate
∮y/(x^2+y^2)dx-x/(x^2+y^2)dy
C
Here C is a square wuth vertices(1,-1),(1,1),(-1,1)and(-1,-1)in R^2
(c)Evaluate
∮y/(x^2+y^2)dx-x/(x^2+y^2)dy
r
Here r is a simple closed counterclockwise in which the enclosed region NOT containing the origin (0,0) in R^2
(d)Evaluate
∫∫ F(x,y,z).ndp
Sr
Here F(x,y,z)=(xi+yj+zk)/p^3 with p=√(x^2+y^2+z^2),Sr is the sphere x^2+y^2+z^2=r^2 of radius r and the outward unit normal n to Sr