93039

呢題應該要點IN我淨係計左呢埋野
之後就諗唔到啦....

alexcheung2009

Wait your final step is inaccurate

Remind that when x=a,u=;x=0,u=a

Therefore it should be
I =  - integrate (0 to a) f(a-u)/[f(a-u)+f(u)] du
  = integrate (a to 0) f(a-u)/[f(a-u)+f(u)] du
= integrate (a to 0) f(a-x)/[f(a-x)+f(x)] dx [dummy variable]

RECALL Definition
I =integrate (a to 0) f(x)/[f(x)+f(a-x)] dx
= integrate (a to 0) f(a-x)/[f(a-x)+f(x)] dx [proven]

Therefore
2I= integrate (a to 0) f(x)/[f(x)+f(a-x)] dx + integrate (a to 0) f(a-x)/[f(a-x)+f(x)] dx
   = integrate (a to 0) [f(x)+f(a-x)]/[f(x)+f(a-x)] dx
   = integrate (a to 0) dx = a
So I = a/2

[ 本帖最後由 alexcheung2009 於 2014-3-29 12:38 AM 編輯 ]

桃子

Find the sum of the first line and last line, then divide by 2.

[ 本帖最後由 桃子 於 2014-3-29 12:59 AM 編輯 ]

93039

引用:

原帖由 alexcheung2009 於 2014-3-29 12:30 AM 發表
Wait your final step is inaccurate

Remind that when x=a,u=;x=0,u=a

Therefore it should be
I =  - integrate (0 to a) f(a-u)/[f(a-u)+f(u)] du
  = integrate (a to 0) f(a-u)/[f(a-u)+f(u)] du
=  ...

點解當個負搖冇左之後會掉轉既?

alexcheung2009

this is one of the rules of integration
See
http://en.wikipedia.org/wiki/Integral
Press Ctrl+F to search for ''
Reversing limits of integration. If a > b then define
''

93039

原來係咁
THANKS但我唔係好明點解A係大過0

[ 本帖最後由 93039 於 2014-3-29 01:02 AM 編輯 ]

桃子

引用:

原帖由 93039 於 2014-3-29 12:47 AM 發表
點解當個負搖冇左之後會掉轉既?

Let F(x) be the antiderivative.

The definite integral from a to b is F(b)-F(a).
The definite integral from b to a is F(a)-F(b).

Hence, the sign reverse.

桃子

引用:

原帖由 93039 於 2014-3-29 12:53 AM 發表
原來係咁
THANKS但我唔係好明點解A係大過0

The expression also holds for a<0 and a=0.
The case a=0 should be trivial.

93039

引用:

原帖由 alexcheung2009 於 2014-3-29 12:30 AM 發表
Wait your final step is inaccurate

Remind that when x=a,u=;x=0,u=a

Therefore it should be
I =  - integrate (0 to a) f(a-u)/[f(a-u)+f(u)] du
  = integrate (a to 0) f(a-u)/[f(a-u)+f(u)] du
= integrate (a to 0) f(a-x)/[f(a-x)+f(x)] dx [dummy variable]

點解可以轉返做X既

桃子

x is a dummy variable.

93039

但題目LET左u=a-x
點解唔係咁樣計既
同埋咩係DUMMY VARIABLE...

[ 本帖最後由 93039 於 2014-3-29 03:10 PM 編輯 ]

MrBenwong

function 入面舊野就係dummy variable

e.g.

假設有f(x) = x^2

咁f(a) = a^2     f(z) = z^2
入面舊野無影響 --> 個VARIABLE唔係重點

93039

o...
明白哂