Find Area of Curved Space | 2D, Positive Geometry

[Niles]

[SOLVED] Area of curved space

Homework Statement

I have the following metric, which describes a 2D, positive curved space with flat geometry (ie. a sphere): $$<br /> ds^2\,=\,dr^2\,+\,R^2 \sin ^2 (r/R)d\theta ^2 <br />$$

Here ds is the distance between two points (r, theta) and (r + dr, theta + dtheta), R is the radius of the sphere.

I want to find the area of the sphere using this metric.

The Attempt at a Solution

Using the metric, I have found the circumference of the sphere to be 2*Pi*R (big surprise). Now I want sum up "all the circumferences" on the sphere. Is that possible?

 

[Niles]

I solved it!

When finding the circumference, I must find all the other circumferences, and r is in the interval [0; Pi*R/2]. Just multiply with 2 in the end (since we only found the first half), and you're set!