# Math Help - Integration ln(x)

1. ## Integration ln(x)

I am new to this forum. Can anyone help me to integrate ln(x) over a circle having radius r. X is the distance from the center to the point, like r. I appreciate any help!

2. Originally Posted by fanta
I am new to this forum. Can anyone help me to integrate ln(x) over a circle having radius r. X is the distance from the center to the point, like r. I appreciate any help!
Use polars $(r, \theta)$, then you seem to be asking for:

$\int_{\theta=0}^{2 \pi} \ln(r) d\theta = 2 \pi \ln(r)$

RonL

3. Thanks CaptainBlack! I am asked to find the average of ln(x) over a circle having radius r. x is the radial distance starting from center to any point. Can the answer 2pi*ln(r) be the answer? thanks for your help.

4. Originally Posted by fanta
Thanks CaptainBlack! I am asked to find the average of ln(x) over a circle having radius r. x is the radial distance starting from center to any point. Can the answer 2pi*ln(r) be the answer? thanks for your help.
No its not the average, the average is just $\ln(r)$ as this is
a constant on the circle.

RonL

5. Thanks again CaptainBlack! In the case of ln(x) --- x is a variable depending upon the distance from the center that is 0<= x <= r. Where r is the radius of the circle. I want to find the average ln(x) over the whole area of the circle for example at the curcumferance ln(x) = ln(r) at the center ln(x) = ln(0)-undefined? How can I get the average ln(x) for the entire circle -- sorry for the misunderstanding and thanks for your help.

6. Originally Posted by fanta
Thanks again CaptainBlack! In the case of ln(x) --- x is a variable depending upon the distance from the center that is 0<= x <= r. Where r is the radius of the circle. I want to find the average ln(x) over the whole area of the circle for example at the curcumferance ln(x) = ln(r) at the center ln(x) = ln(0)-undefined? How can I get the average ln(x) for the entire circle -- sorry for the misunderstanding and thanks for your help.
Ahh.. slightly different problem.

Now we need:

$
\mu = (1/(\pi r^2) \int_{R=0}^r \int_{\theta=0}^{2 \pi} R\ \ln(R)\ d \theta dR
$

which is the integral of $\ln(R)$ over the circle of radius $r$ in polars (where the area element is $R\ dR d\theta$) divited by the area of the circle.

RonL

7. is there a solution for this integral then?

8. Originally Posted by fanta
is there a solution for this integral then?
O-yes, but rather than slog through it by hand, I will ues machine assistance
for the final step. First change the order of integration and integrate out $\theta$

..... $=\frac{2}{r^2} \int_{R=0}^r R\ \ln(R) \ dR$

Now using QuickMath this becomes:

..... $= \ln(r)-1/2$

9. CaptainBlack . Thanks so much that's what I want!

10. CaptainBlack! I am quite gratful for your help and satisfied with your answer; however, I have got the same question but the circle diplaced from the origin by a distance x. The center of the circle is displaced by a distance x from the origion where the this distance, x is larger than the radius of the circle.

11. Originally Posted by fanta
CaptainBlack! I am quite gratful for your help and satisfied with your answer; however, I have got the same question but the circle diplaced from the origin by a distance x. The center of the circle is displaced by a distance x from the origion where the this distance, x is larger than the radius of the circle.
It is not clear to me what you want integrated over which region, as you
seem to be using x for two different things in this diagram.

RonL

12. Thanks for your reply! I am interested to find the average lan(x) in the circle. At any point inside the circle the function varies as lan(x) where x is the distance from the origin to any point in the circle. I want to find the average lan(x) inside the circle.[In the previous case we find the average lan(r) where the center of the circle is at the origin.]

13. I described it here graphically.

I appreciate your help CaptainBlack

14. I think this can be reduced to this integral Can anyone help me to resolve this problem. where (a,b) is the center of the circle.

15. Originally Posted by fanta
I think this can be reduced to this integral Can anyone help me to resolve this problem. where (a,b) is the center of the circle.
Over what region are you integrating? (Expressed in Polar Coordinates).

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