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）︰99/12/21 考试时限（分钟）：30分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : ˙每题十分 ˙请详述计算过程，无计算过程的答案不予计分 1. Find the length of the curve. y = ㏑[sec(x)] , 0 ≦ x ≦ π/4 (10%) d ㏑[sec(x)] tan(x)sec(x) y' = ─────── = ─────── = tan(x) dx sec(x) π/4 ＿＿＿＿＿＿＿＿ π/4 ∫ √1 + [tan(x)]^2 dx = ∫ sec(x) dx 0 0 π/4 sec(x) + tan(x) = ∫ sec(x) * ─────────dx 0 sec(x) + tan(x) ┌ ┐π/4 = │ ㏑│sec(x) + tan(x)│ │ └ ┘0 = ㏑[√(2) + 1] - ㏑(1) = ㏑[√(2) + 1] ＃ 2. Find the area of the resulting surface. y = x^2 from (1,1) to (2,4) is rotated about the y-axis. (10%) d(x^2) y' = ──── = 2x dx 2 ＿＿＿＿＿＿ 2 ＿＿＿＿＿ ∫ 2πx √1 + (2x)^2 dx = 2π * ∫ √4x^2 + 1 dx 1 1 ┌ 3/2 ┐2 = │ (π/6)(4x^2 + 1) │ └ ┘1 ┌ 3/2 3/2 ┐ = (π/6)│ 17 - 5 │ └ ┘ ＃ t+z 3. Solve the differential equatation. dz/dt + e = 0 (10%) t z dz/dt = -(e * e ) -z t => ∫(-e ) dz = ∫(e ) dt => e^(-z) = e^(t) + C => -z = ㏑│e^(t) + C│ │ 1 │ => z = ㏑│──────│ │ e^(t) + C │ ＃ 4. Solve the differential equation. xy' = y + (x^2)*sin(x) , y(π) = 0 (10%) y' - y/x = x*sin(x) ∫(-1/x) dx -㏑|x| ∴ I(x) = e = e = 1/x 两边同乘 I(x)： y' y => ──- ───= sin(x) x x^2 => (y/x)' = sin(x) => ∫ d(y/x) = ∫ sin(x) dx => y/x = -cos(x) + C => y = (-x)*cos(x) + C*x y(π) = π + π*C = 0 => C = -1 ∴ y = (-x)*cos(x) - x ＃ --

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