课程名称︰微积分甲下
课程性质︰数学系大一必带
课程教师︰陈荣凯
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2014/05/08
考试时限（分钟）：30
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
The total is 25 points.
2 2
(1) (12 pts) Consider a torus T obtained by rotating a circle ( x - b ) + z =
2
a with 0 < a < b around z-axis. Find a parametric representation and
evaluate the area of the torus T.
(2) (13 pts) Let C be the line segment connecting P = ( x , y ) to P = ( x ,
1 1 1 2 2
y ). Show that ∫ x dy - y dx = x y - x y . Moreover, given a polygon
2 C 1 2 2 1
with vertices P , ..., P in counterclockwise order. Write P = ( x , y ),
1 n i i i
1
then the area of the polygon is - [ ( x y - x y ) + ( x y - x y ) + ...
2 1 2 2 1 2 3 3 2
+ ( x y - x y ) ].
n 1 1 n