微積分與統計的不定積分問題

Q1
Part a
Let f(x) = x/(1+x^2)
Put x=0,
f(0) = 0 / (1+0) =0
so the curve must pass through (0,0), i.e. 原點

Part b
Area = Int [x ranges from 0 to 2] {x/(1+x^2)}dx
= Int [x ranges from 0 to 2] {dx^2/2/(1+x^2)}
=Int [x ranges from 0 to 2] 1/2 * d(1+x^2)/(1+x^2)
=1/2 *[ ln(1+x^2)](0,2)
= 1/2 *{ln 5 - ln1)
=ln5 / 2

Q2
Part a
Area between the curve and x-axis
= Int [x ranges from 0 to a] (x^3)dx
= Int [x ranges from 0 to a] (x^3)dx
= x^4 / 4 (0, a)
=a^4 / 4 - 0
=a^4 / 4

Area between the curve and y-axis
= a*b - a^4/4
since b= a^3
hence
Area between the curve and y-axis
= a * a^3 - a^4 / 4
= 3/4 * a^4

Part b
Area between the curve and x-axis/ Area between the curve and y-axis
= a^4 / 4 / ( 3/4 * a^4)
=1/3

1(a) 0/(1 + 0^2) = 0 因此y = x/(1 + x^2) 通過原點

(b) ∫ x/(1 + x^2) dx (0 -> 2)

= (1/2) ∫ 1/(1 + x^2) d(1 + x^2) (0 -> 2)

= (1/2)ln(1 + x^2) | 0,2

= ln5/2

2(a) 垂直於x軸的面積

= ∫ x^3 dx (0->a)

= a^4/4

垂直於y軸的面積

= ∫ y^(1/3) dy (0->b)

= 3b^(4/3)/4

= 3a^4/4

(b) 區域1與區域2的面積比為1：3

= a^4/4 : 3a^4/4

= 1 :3