>I have studied only analysis and from what I know calculus is supposed to be analysis without proofs.
In my opinion, this is wrong.
"Calculus" ("differential and integral calculus", "calcul différentiel et intégral" or "Differential- und Integralrechnung") is an older name for the mathematical discipline that came to be known as analysis. To the extent there is a distinction between these two terms, it is that calculus deals more specifically with those aspects related to the processes of differentiation and integration.
In English-speaking countries, within the context of education, the use of the word "analysis" never caught on other than in advanced university courses, while in continental Europe, the reverse occurred, with the term "analysis" displacing the older usage of "differential and integral calculus".
The books "Calculus" by Apostol and Spivak and the book by Courant originally published in 1927 under the title "Vorlesungen über Differential- und Integralrechnung" would be described as nothing but rigorous, with full proofs. There are courses taught in American universities from these books, also under the title "Calculus." On the other hand, the subject "Analyse" is taught in school in many European countries without many proofs or in any case at a less rigorous level than the books above. No doubt first-year courses are far more rigorous on the whole in Germany than they are in the U.S., but this has nothing to do with an inherent distinction between the words "calculus" and "analysis."