the average distance from any point inside a circle radius 10 to an arbitrary point on the edge of the circle?

Question

I already solved this if it is the average distance from orgin to any poinit in the circle but am not sure how to solve for a point on the circle--btw from center to any point is an avg. of 6 and change so im assumiing the correct answer for my question will be in the 10-12 range! thanx.. this involves a double integral in polar cord.

Answer 1

Try to turn the problem around to make it easier to integrate. The following is equivalent and has the same answer:

On the circle x^2 +(y-10)^ =10^2, find the average distance from the point (0,0) on the circle to any point inside the circle. (*note that the point on the circle has been oriented to coincide on the origin for simplicity).

Answer 2

particular. while the priority says that a element is chosen at random, that's attempting to enable you know that the possibility of choosing a element interior a definite area is proportional to the component of that area. (that's no longer what it extremely says -- strictly speaking, it desires to assert "uniformly at random" in the different case you haven't any longer have been given adequate information to sparkling up the priority -- yet we are in a position to enable the errors bypass because of the fact that's glaring what's meant.) the component of the smaller circle is pi and the component of the extra beneficial circle is 4pi. subsequently the possibility of choosing a element interior the smaller circle is (pi / 4pi) = a million/4. Intuitively, you could think of that the smaller circle fills up a million/4 of the extra beneficial circle, so in case you % factors interior the huge circle, a million/4 of them would be interior the smaller circle. (that's no longer a appropriate mathematical argument, however.)