Expert: Paul Klarreich - 4/5/2008

Question
Hi Paul,

I have a question about definite integrals. Here's the problem:

Assuming that [integration from 0 to 1] f(x)dx = 1, [integration from 0 to 2]
f(x)dx = 4, [integration from 1 to 4] f(x)dx = 8, calculate:

[integration from 4 to 2] f(x)dx = ??

Thank you so much for your help!

Answer
Questioner:   Ashley
Category:  Calculus
Private:  No
 
Subject:  Integration
Question:  Hi Paul,

I have a question about definite integrals. Here's the problem:

Assuming that [integration from 0 to 1] f(x)dx = 1, [integration from 0 to 2]
f(x)dx = 4, [integration from 1 to 4] f(x)dx = 8, calculate:

[integration from 4 to 2] f(x)dx = ??

Thank you so much for your help!
 

{1
|   f(x) dx = 1
}0

{2
|   f(x) dx = 4
}0

{4
|   f(x) dx = 8
}1

Rule:

{b  {c   {c
|  +|  = |
}a  }b   }a

I shall abbreviate:  a::b shall mean the integral from a to b of f(x).
{My fingers tire easily these days.}


So that last says:   a::b + b::c = a::c

Now  0::1 + 1::2 = 0::2, and since 0::1 = 1 and  0::2 = 4, it must be
that  1::2 = 3.

AND:  1::2 + 2::4 = 1::4, and since 1::2 = 3 and 1::4 = 8, it must be
that  2::4 = 5.

Also there is a rule:  a::b = - b::a.

So 4::2 = -5.

That's it.