Expert: Paul Klarreich - 3/13/2006

Question
1 kyle
2 calculus and the substitution rule
3 ?


if f is continuous and s from 0 to 9 f(x)dx=4, find s from 0 to 3 xf(xsquared)dx

this is difficult to write in this media
s = integral sign
0 to 9 means 0 is subscript and 9 is above the s like a power would be.

thanks I hope you understand what I am asking.


Answer
Hi, kyle,
You wrote:
Subject:  integral calculus and the substitution rule
Question:  1 kyle
OK, OK,  I get your name from the system, as long as you provide it.

2 calculus and the substitution rule
[And the subject line is sent, too, so you just have to put it there.]

3 ? <<-- This is a short description of the subject you are studying right now, and the point you have reached.  In this case, you are probably studying Calculus II and the section heading in the text says 'Properties of the Definite Integral.'

if f is continuous and s from 0 to 9 f(x)dx=4, find s from 0 to 3 xf(xsquared)dx

this is difficult to write in this media

>>> I know. See below.

s = integral sign
0 to 9 means 0 is subscript and 9 is above the s like a power would be.

thanks I hope you understand what I am asking.
---------------------
Here's how to write your question:(using a fixed font, always)

If:
{x=9
|    f(x) dx = 4
}x=0

what is:

{x=3
|    x f(x^2) dx
}x=0

Now, about answering it, the key to REALLY understanding a definite integral is to realize that:

{x=9
|    f(x) dx
}x=0

and

{t=9
|    f(t) dt
}t=0

and

{w=9
|    f(w) dw
}w=0

are all (not sort-of the same, not the same kind of thing, but) EXACTLY THE SAME THING.  The variable of integration (x,t,w, whatever) is a 'dummy variable' -- when the calculation is completed, it disappears.

Now for

{x=3
|    x f(x^2) dx
}x=0

Do a routine substitution:

w = x^2
dw = 2x dx, so  
dw/2 = x dx, which is in the integral.

Also, when changing the variable of integration, you will want to change to equivalent boundaries:

x = 0  gives  w = 0
x = 3  gives  w = 9

So the integral becomes:

1 {w=9
-- |    f(w) dw  =
2 }w=0

1 {x=9             1
-- |    f(x) dx  = --- (4) = 2
2 }x=0             2

And do not worry about the presence of x in the last one.  Remember, it is a dummy variable.