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!
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:
$\displaystyle
\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 $\displaystyle \ln(R)$ over the circle of radius $\displaystyle r$ in polars (where the area element is $\displaystyle R\ dR d\theta$) divited by the area of the circle.
RonL
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 $\displaystyle \theta$
..... $\displaystyle =\frac{2}{r^2} \int_{R=0}^r R\ \ln(R) \ dR$
Now using QuickMath this becomes:
..... $\displaystyle = \ln(r)-1/2$
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.
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.]