# Help with Multivariable Calculus

Find the average area of an inscribed triangle in the unit circle. Assume that each vertex of the triangle is equally likely to be at any point of the unit circle and that the location of one vertex does not affect the likelihood the location of another in any way. (Note that, as seen in Problem set 4, the maximum area is achieved by the equilateral triangle, which has
side length √3 and area 3√3/4. How does the maximum compare to the average?)
Hint: in order to reduce the problem to the calculation of a double integral, place one of the vertices of the triangle at(1,0),and use the polar angles θ1 and θ2 of the two other vertices as variables. What is the region of integration?

Thanks

http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-02Fall-2007/Assignments/ps7.pdf [Broken]

(I'm not at MIT, I'm taking the class online, and this problem REALLY irritated me)

Last edited by a moderator:

## Answers and Replies

EnumaElish
Science Advisor
Homework Helper
If this is HW, it should have been posted under the HW forums.

Do you have a specific question? The region of integration is given by the domain of the integration variables, θ1 and θ2. Picture one side of the triangle making a sweep inside the unit circle (sort of like a radar monitor) -- what is the minimum angle? What is the max. angle?

If this is HW, it should have been posted under the HW forums.

Do you have a specific question? The region of integration is given by the domain of the integration variables, θ1 and θ2. Picture one side of the triangle making a sweep inside the unit circle (sort of like a radar monitor) -- what is the minimum angle? What is the max. angle?

It isn't homework... as I'm not in a class. But it is a "problem," so I actually reposted it in the HW section.