Why "integralis" over "summatorius"? It is written that Johann Bernoulli suggested to Leibniz that he (Leibniz) change the name of his calculus from "calculus summatorius" to "calculus integralis", but I cannot find their correspondence wherein Bernoulli explains why he thinks "integralis" is preferable to "summatorius." Can someone enlighten me? Thank you. 

I have no exact answer to your question, but I can provide you with some background. In late medieval Italian the adjective "integrale" was already used; its meaning is "total, entire". It is derived from the Latin word "integralis", which belongs to non-classical Latin (6th century A.C., so definitly non-classical): it appears in some comments to Cicerone's works. The Latin word "integer" is classical, instead. It has the same root of "tangere" (to touch) and "tangente" (yes, tangent!) and it means..."integer" or "pure/untouched".
Probably J. Bernoulli used the word "integralis" in 1690 to specify the property of "totality" or "completeness" of the operation, as the integral computes the "whole" area under a given curve.
Euler uses "integralis" in his "Institutiones calculi integralis" as an adjective.
It is funny to note that in the 18th century Luigi Guido Grandi and Maria Gaetana Agnesi discussed about "calcolo integrale" (i.e. Integral Calculus) using the word "integrale" as an adjective, while Riccati uses "integrale" as a noun.
