Aren't they the same thing?
Depends who you ask. Some people take analysis to be the rigorous foundation of what they call calculus.
It might be beneficial to think of it interms of
Theory and Prcactice.
One defines an integral and proves things about it, the other demonstrates how to acutally integrate honest to goodness things like sin(x).
Oh, and there is also the fact that some people might not use one of the names, so you might want to view it as calculus (aka analysis) so people know what it is about.
sometimes you even get to hear 'advanced calculus' as a term synonymous with analysis.
Unfortunately, "advanced calculus" is also used (particularly by engineers) to mean more advanced techniques for solving problems: differential equations, special functions, etc.
No, not really, we can analyze a calculus (on the teeth, in the kidnees,...) , but we can't calculate an analysis.
In a calculus course, epsilon-delta proofs are encountered only in the most formal sense, and are usually regarded as dreaded "rites of passage," rarely understood in lecture, suffered through in homework, and promptly forgotten after the exam; whereas in an analysis course, they are like unto water, or even air: you cannot survive without them, no, not even for a few minutes.
Separate names with a comma.