课程名称︰微积分甲上
课程性质︰必修
课程教师︰颜文明(陈宏代课)
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/08/04
考试时限（分钟）：90
试题 :
1. (10%) Evaluate ∫_0^1 x(arctan(x))^2 dx
2. Consider the differential equation xy' = y + xe^(y/x)
(a) (5%) Rewrite it into a new differential equation in terms of v and x by
making the change of variable v = y/x
(b) (10%) Solve the equation obtained in (a) and deduce the solution of
xy' + xe^(y/x)
3. (15%) Solve the initial value problem
{ (sec(x)y' + y = tan(x)e^(cos(x) - sin(x)), 0 ≦x ≦π/2,
y(0) = 0
4. Let the curve C defined by x = cos(t)^3, y = sin(t)^3, 0 ≦t ≦π/2
(a) (10%) Find dy/dx and (d^2)y/dx of the curve C.
(Express the answers in terms of t.)
(b) (10%) Find the area of the region enclosed by the curve C, x-axis, and
y-axis.
(c) (10%) Find the length of the curve C.
(d) (10%) Find the surface area generated by rotating the curve C about the
y-axis.
5. (15%) Find the length of the curve in polar coordinates:
r = (1+sin(2θ))^(1/2), 0 ≦θ≦2π.
6. (a) (5%) Find all intersection points of the curves r = 2 + cos(2θ) and
r = 2 + sin(θ).
(b) (10%) Find the area of the region that lies inside the curve
r = 2 + cos(2θ) but outside the curve r = 2 + sin(θ)