課程名稱︰微積分甲上 課程性質︰必修 課程教師︰顏文明(陳宏代課) 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/08/04 考試時限（分鐘）：90 試題 : 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) --

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