I thin this video capture the essence of calculus.

"We have a geometric concept, and we want an algebraic expression for it".

It could be: we have XYZ concept, and  want and algebraic expression for it.

An XYZ calculus (such as Tensor calculus in case of Geometry, differential calculus in case of algebra, Relational Calculus in case of Relational Algebra, Lambda Calculus in case of Function algebra...)  is  used to write an algebraic expression which is then solved with algebraic laws to reason.  

The method seems to always be "Write down the correct identity"...  Struggling to find how this applies to Lambda Calculus..