Original Post
Hello! I am working on atmospheric scattering and therefore I have to calculate the optical depth along a given path through the atmosphere. The integral I have to solve is this one: 
Pa und Pb are Points in R^3 and I have to solve the integral along the path from Pa to Pb. H0 is just a constant. So obviously this is a line integral (at least I hope so), which can be solved this way: 
This gamma (Im gonna call it r) is the parameterization of the path, so my r is: r: [a,b] -> R^3. r(t) (t is in [a,b]) gives me a point on the path from Pa to Pb, with r(a) = Pa and r(b) = Pb. Now I have the problem that the result of r doesnt fit to the input of f, because f needs a height and not a point in R^3. Therefore I slightly change f (and now call it f2): 
f2 expects a point in R^3 and calculates the height of the point P itself. Now I just need the parameterization r of the path. Two solutions came into my mind (I had to write plus instead of using the plus symbol cause the forum doesnt show the plus sign): 1) r: [0,1]->R^3, t -> Pa plus t * (Pb - Pa) with r'(t) = LengthFromPaToPb = Length 2) r: [0, N]->R^3, t -> Pa plus t * (Pb - Pa) / N with r'(t) = LengthFromPaToPb / N = sampleLength If I pick parameterization 2) I end with this formula: 
I numerically solve this integral with the trapezium rule. My problem is, that the solutions I get with the trapezium rule are really big (up to 14000). Maybe its because the length of my paths between Pa and Pb are long (about 80000meters) and thus my Length and sampleLength are also very big. I cant believe that values around 14000 are right. So Im asking you if I made a mistake. Im very happy for every idea!
Pa und Pb are Points in R^3 and I have to solve the integral along the path from Pa to Pb. H0 is just a constant. So obviously this is a line integral (at least I hope so), which can be solved this way: This gamma (Im gonna call it r) is the parameterization of the path, so my r is: r: [a,b] -> R^3. r(t) (t is in [a,b]) gives me a point on the path from Pa to Pb, with r(a) = Pa and r(b) = Pb. Now I have the problem that the result of r doesnt fit to the input of f, because f needs a height and not a point in R^3. Therefore I slightly change f (and now call it f2): f2 expects a point in R^3 and calculates the height of the point P itself. Now I just need the parameterization r of the path. Two solutions came into my mind (I had to write plus instead of using the plus symbol cause the forum doesnt show the plus sign): 1) r: [0,1]->R^3, t -> Pa plus t * (Pb - Pa) with r'(t) = LengthFromPaToPb = Length 2) r: [0, N]->R^3, t -> Pa plus t * (Pb - Pa) / N with r'(t) = LengthFromPaToPb / N = sampleLength If I pick parameterization 2) I end with this formula: I numerically solve this integral with the trapezium rule. My problem is, that the solutions I get with the trapezium rule are really big (up to 14000). Maybe its because the length of my paths between Pa and Pb are long (about 80000meters) and thus my Length and sampleLength are also very big. I cant believe that values around 14000 are right. So Im asking you if I made a mistake. Im very happy for every idea!