# Quick question about finite area / infinite area

• April 20th 2008, 03:24 PM
larson
Quick question about finite area / infinite area
Here's a question I have...

Let S be the region between the graphs $y = \frac {1}{x}$ and $y = \frac {1}{\sqrt {x}}$ between x = 0 and x = 1. Does S have finite area? Justify your answer.

Now, so do I first do this... $\int_{0}^{1} \frac{1}{x}$ and I figured out that this ends up diverging to $\infty$. So does this mean that S has an infinite amount of area? Please tell me if I'm doing this correctly. Thanks.
• April 20th 2008, 07:36 PM
topsquark
Quote:

Originally Posted by larson
Here's a question I have...

Let S be the region between the graphs $y = \frac {1}{x}$ and $y = \frac {1}{\sqrt {x}}$ between x = 0 and x = 1. Does S have finite area? Justify your answer.

Now, so do I first do this... $\int_{0}^{1} \frac{1}{x}$ and I figured out that this ends up diverging to $\infty$. So does this mean that S has an infinite amount of area? Please tell me if I'm doing this correctly. Thanks.

You are taking
$\int_0^1 \frac{1}{\sqrt{x}}~dx$
from this infinite area.

Are you subtracting an infinte area from an infinite area? This one's a toughy. What you are going to have to do is
$S = \int_0^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx$

$S = \lim_{a \to 0} \int_a^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx$

And express that as a limit that has the indeterminate form of $\infty - \infty$. Do you know how to find this limit? (You'll probably end up using L'Hopital's rule at some point to give you a pointer as to where to look up how to do this.)

-Dan
• April 20th 2008, 07:38 PM
Mathstud28
Quote:

Originally Posted by topsquark
You are taking
$\int_0^1 \frac{1}{\sqrt{x}}~dx$
from this infinite area.

Are you subtracting an infinte area from an infinite area? This one's a toughy. What you are going to have to do is
$S = \int_0^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx$

$S = \lim_{a \to 0} \int_a^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx$

And express that as a limit that has the indeterminate form of $\infty - \infty$. Do you know how to find this limit? (You'll probably end up using L'Hopital's rule at some point to give you a pointer as to where to look up how to do this.)

-Dan

I think I have turned the tide against L'hopitals! Hooray! (Rock)
• April 20th 2008, 07:59 PM
topsquark
Quote:

Originally Posted by Mathstud28
I think I have turned the tide against L'hopitals! Hooray! (Rock)

No, I've always been a bad boy about that... (Mooning) (as I'm not nearly clever enough to not use it!)

-Dan
• April 20th 2008, 08:03 PM
Mathstud28
Quote:

Originally Posted by topsquark
No, I've always been a bad boy about that... (Mooning) (as I'm not nearly clever enough to not use it!)

-Dan