””微積分””~快幫幫我<<特急!!!>>

Question

這裡有三題關於面積體積的問題

拜託各位高手幫忙~~ = = 我有解答~~

*the area of the surface generated by revolving the curve x=y得1/2次方

0=< y =< 2 about the y-axis is 13拍/6 ?




*the area of the rigion enclosed by y^2- 4x = 2 and 4x - y = 0 is 9/8 ??




*let D be the region enclosed by the triangle ABC where A = (1,0) ,B=

(2.1) , C = (1,1) .then the volume of the solid generated by revolving D

about the y-axis is 4拍/3 ??




拜託過程~~~謝謝囉~!! 感激不盡

請問你是回答第幾題??
你說的觀念我懂!!
但還是沒幫我解........

Answer 1

1. 對 y 軸旋轉
表面積 = π∫02 x√{1 + [x']2} dy
= π∫02 (√y)√{1 + [1/(2√y)]2} dy
= π∫20 [√(4y +1)]/2 dy
= (π/2)* {(1/6)*(4y+1)(3/2) | 20}
= (π/12)*(27-1)
= 13π/6
2. y2- 4x = 2 和 4x - y = 0 的交點 (1/2, 2), (-1/4, -1)
面積 = ∫-12 (x1 - x2) dy
=∫-12 [y/4 - (y2-2)/4] dy
= (1/4)*∫-12 [-y2 + y + 2] dy
= (1/4)* [-y3/3 + y2/2 + 2y | 2-1}
= (1/4)* (-8/3 + 2 + 4 - 1/3 - 1/2 + 2)
= 9/8
3. AB 直線方程: x = 1; AC 直線方程: x = y + 1
體積 = π∫01 (x12 - x22) dy
= π∫01 [(y+1)2 -12] dy
= π∫01 [y2 + 2y] dy
= π[y3/3 + y2 | 10]
= 4π/3
如果有問題, 請來函討論. 不然, 我可能會錯失你再補充的疑點.

Answer 2

By definition of Limit,given any ε>0,choose M€R with M>0,such


that for x≧M,we have:| f(x)-L|<ε


Choose δ>1/M


then if 1/y≧M,implies | f(1/y)-L|<ε


so if 0|f(1/y)-L|<ε


Hence lim(y->0+)f(1/y)=L
建議你重讀觀念

微分在基本的應用範疇是在求函數的切線斜率(微一次)

在微分第二次是求出函數斜率所形成的函數(即導函數)的切線斜率

應用在函數凹口方向的決定