作者rod24574575 (天然呆)
看板NTU-Exam
标题[试题] 99上 薛克民 微积分甲上 第六次小考
时间Fri Dec 10 01:38:09 2010
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）︰99/11/29
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
€ = 属於
Ｒ = 实数
1. The set { (x,y)€Ｒ^2 │ y^2 = x^2 - x^4 , 0 ≦ x ≦ 1 } enclose a region
＿＿＿＿＿
in Ｒ^2 . Find the area of this region. (Hint: y = ±√(x^2 - ^4) (10%)
2. Find the volume of the solid bounded by the surface 4y^2 + z^2 = 1 - ^2
and z = 0. As shown in figure, the base of this solid is the region
bounded by x^2 + 4y^2 = 1. Cross sections perpendicular to x-axis is half
ellipse(椭圆) with endpoint of major axis(长轴端点) lying on the curve
z = √(1-x^2), and minor axis(短轴) in the base. (10%)
┌ x ┐2 ┌ y ┐2
(Hint: the area of ellipse │
── │ + │
── │ = 1 is abπ)
└ a ┘ └ b ┘
3. Let R be the region lying between y = [cos(x)]/x , x = 0 and y = 0. Find
the volume of the solid generated by rotating R about the y-axis. (10%)
4. Let ╭ sin(x)
│ ───── , if x ≠ 0
f(x) = ┤ √(|x|)
│
│ 0 , if x ＝ 0
╰
(a) Show that f is continous on [-π,π]. (5%)
(b) Find the average value of f on [-π,π]. What is the number c€(-π,π)
such that f(c) = the average value of f on [-π,π]? (5%)
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