Finding the geodesic function for scalar * function= scalar

[metric tensor]

In this expression the junk on the left is a scalar. The stuff before the integral is another scalar. The integral is a time-like curve between x₁ and x₂ and at imagine f_gf(x₁) is a lower left corner of the rectangle and f_gf(x₂) is the upper right corner and x₂-_x1 is the length of the base of the rectangle. I know the geodesic is the shortest distance curve paramaerized by this on the condition that the area (the integral) times the scalar₁ = scalar₂. The solution isn't a strait line but could be some curved function. How do I find the function f_gf ?
Do i need to find the 2x2 metric tensor for the 2-d paramaterization curve and solve a pair of differential equations like here ?

http://count.ucsc.edu/~rmont/classes/121A/polarChristoffelE_Lag.pdf

 

[member 553083]

Unfortunately, it is not possible to answer this question without knowing more details about the specific problem. However, in general, if you have an integral equation of the form:scalar1 = scalar2*Integral[fgf(x), x1, x2]Then you can solve for fgf by taking the integral of both sides and rearranging the equation:fgf(x) = (scalar1/scalar2)*1/(x2-x1)This should give you the function fgf that you need.