※ 引述《gghh711 (lara)》之銘言：
: 1.試求x=y^2,x=4為界的區域,繞著x=-1旋轉所包圍的體積
: Ans:1088π/15
: 2.試求y=4x,y=x/2,y=1/x與x>0 所包圍的面積
: Ans:3ln2/2
將 y = 4x , y = x/2 , y = 1/x , x > 0 畫在xy平面得
y = 4x
=> (x,y) = (0,0)
y = x/2
y = 4x 1
=> (x,y) = (--- , 2)
y = 1/x 2
y = x/2 1
=> (x,y) = (√2 , ----)
y = 1/x √2
1/2 x √2 1 x
所求面積 = ∫ 4x - --- dx + ∫ --- - --- dx
0 2 1/2 x 2
x^2 |1/2 x^2 |√2
= (2)(x^2) - --- | + ln|x| - --- |
4 |0 4 |1/2
1 1 ln2 1 1
= (--- - ----) + (--- - ---) - (-ln2 - ----)
2 16 2 2 16
ln2 3
= --- + ln2 = (---)(ln2)
2 2
: 3.Consider the plane region R(t)={(x,y)│0<=x<=t,0<=y<=1/(1+x^2)},let
: V(t)be the volume of the solid obtained by revolving R(t)about the x-axis
: ,then lim V(t)=?
: t=>無窮
: Ans:π^2/4
: 感謝
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