課程名稱︰微積分甲上
課程性質︰必修
課程教師︰薛克民
開課學院：電資學院、工學院、管理學院
開課系所︰電機系、資工系、材料系、資管系
考試日期（年月日）︰99/11/29
考試時限（分鐘）：30分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
˙每題十分
˙請詳述計算過程，無計算過程的答案不予計分
€ = 屬於
Ｒ = 實數
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(橢圓) with endpoint of major axis(長軸端點) lying on the curve
z = √(1-x^2), and minor axis(短軸) 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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