Expert: Paul Klarreich - 2/14/2006

Question
Goodday,
Please how am i supposed to find the area below the curve y = 1/x from x = 0 to x = 2.
I'm having problems with the lnx part.
Thanking you in advance.  

Answer
Hi, Pharell,

You wrote:
Subject:  Areas
Question:  Goodday,
Please how am i supposed to find the area below the curve y = 1/x from x = 0 to x = 2.
I'm having problems with the lnx part.
Thanking you in advance.  
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Of course you are having trouble with ln x.  Ln(0) is undefined.  Not only that,

limit(x->0) ln x  does not exist.

WARNING: THE MATERIAL BELOW MAY CONTAIN FRACTIONS AND/OR OTHER MATERIAL INAPPROPRIATE FOR SOME COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

You have an example of something called an 'improper integral'.  That means some definite integral in which one or more of these things happen:

A. The left boundary of the integral is infinite.
B. ..  right ...................................
C. The integrand is undefined at the left boundary.
D. ................................. right ........

In this case, it's C.  Your integral is:


{x=2
|    1/x  dx
}x=0

Now sometimes an improper integral can be computed in spite of the 'problem.'  In your case, you define the integral to be:

        {x=2
lim(a->0)|    1/x  dx
        }x=a

and try to integrate and then compute the limit.  You get:

lim(a->0) [ln 2 - ln a] = ln 2 - lim(a->0) ln a

But unfortunately, that limit does not exist.  As a->0, ln a approaches -infinity.

So of course you will have trouble with it.