课程名称︰微积分甲上 课程性质︰必修 课程教师︰薛克民 开课学院：电资学院、工学院、管理学院 开课系所︰电机系、资工系、材料系、资管系 考试日期（年月日）︰99/12/31 考试时限（分钟）：110分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. y^3 = x^2 defines a curve on xy—plane. (a) Find the arc length of this curve from (-1,1) to (8,4). (b) Find the area of the surface generated by revolving about x-axis this curve from (0,0) to (1,1). 2. (a) Solve the differential equation x y' - [tan(x)]*y = ────── [sin(x)]^2 (b) Solve the differential equation x y' - [tan(x)]*y = ────── * (y^3) [sin(x)]^2 3. Calculate the following integral (a) 1 ∫ ──────dx √(1+e^x) (b) x^3 + x^2 ∫ ───────── dx (x^2 + 2x + 2)^2 4. (a) Find all intersection points of the curve y^2 = x^3 - 2xy and y = mx. (b) Find a parametrization of the curve y^2 = x^3 - 2xy, and find all lines tangent to the curve at (0,0). 5. The curve defined by x^2 + 2xy - y^2 = 1 is a hyperbola(双曲线) (a) Write this curve in polar coordinates, and use it to determine the asymptotes of this curve. (b) Using polar coordinate to compute the area of the region bounded by this hyperbola and a straight line passing through origin θ = c lying between x-axis and the asymptotes in first quadrant(第一象限). (c) What is the area of the region bounded by the hyperbola and its asymptotes in first quadrant? 6. (a) Find the area of the surface generated by revolving about x-axis the curve y = 1/x from x = 1 to x = c for some c > 1. Use it to show that the surface area generated by rotating {(x,y)│y = 1/x, x≧1} is infinite. (b) Is the surface area generated by rotating {(x,y)│y = (1/x)^2, x≧1} finite? Is the surface area generated by rotating {(x,y)│y = (1/x)^3, x≧1} finite? In general, for what n is the surface area generated by rotating {(x,y)│y = (1/x)^n, x≧1} finite? --

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