There is something known as conditioning that does not require reason, but simply programs a living creature nor matter how high or low to a set of responses. It's not a matter of reason or faith, but simply of rewards. Feels good; feels bad.
With all due respect to your professor, an area of any shape can be calculated if proper expressions are integrated. An expression is broken every-diminishing derivatives and an exact area is theoretically possible in a theoretical world. In the real world, the precision is carried to practical limits. So, the answer is -- theoretically -- yes; practically no. And his observation is realistically correct: consistency is much more important for most applications.
Of course one uses integration. That is what the professor and I were discussing.
Exactly what means do you personally use to confirm the exactness of a result arrived at by integration?