# which preposition must be used "through" or "by"

In sentences :

1. We can derive a closed-form expression through integration of X.

2. The closed-form expression can be derived by integration of X.

3. We can derive the optimal solution through optimization of X.

4. The optimal solution can be derived by optimization of X.

Given to here, we are recommended that to use "through" when we have a process ( i.e., an start point + a procedure and a final result). Also, we should use "by" when we want to focus on method.

"through integration" = 478 k

"by integration" = 710 k

"through optimization" = 206 k

"by optimization" = 320 k

Now the question is, in above context ( Mathematics ) which one we should use? We can infer both a method and a process. ( the integration can be complex and you might need to do some complex calculations in multiple stages). However, google demonstrates the usage of "by" is more frequent.

PS : I think if we use through, we say the significant part of the following is the procedure. However, when we use by, we should expect the reader focusing on the used tool. In fact, we induct the procedure is unimportant and superficial. Am I right ?

Thanks.