It's somthing I always want to figure out, when did calculus start to be extended to analysis(I reformulate the question, the previous one"where one can draw a line to distinguish calculus and analysis, or there does not even exist such a line." was quite misleading).
As mentioned a lot in comments, analysis is a much border field than calculus, but the root could be traced back to the calculus in 19th century.
Besides, indeed infinitesimal calculus was proved in non-standard analysis, but it was invented until 1960s I think. And I don't know if it can replace all arguments in the theories developed after $\varepsilon-\delta$-definition and before the invention of non-standard analysis.
I will explain what I understand, please point out my mistakes.
The early stage (Newton and Leibniz)
They used infinitesimal, say $\mathrm{d}(\cdot)$ to describe change such as $\mathrm{d}x$ and $\mathrm{d}y$.
And use
$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y(x+\mathrm{d}x)-y(x)}{\mathrm{d}x}$$
to compute derivatives.
And let $y'$ be a shorthand notation for $\frac{\mathrm{d}y}{\mathrm{d}x}$, they defined integral as sum over infinitesimals
$$\int y' \mathrm{d}x. $$
(I do not know how Newton and Leibniz defined integral. Maybe as $\approx\sum y(x_i)\Delta x$?)
19th century
People started worrying about the precision of infinitesimals. And the ratio of infinitesimals was replaced by limit (the '$\varepsilon-\delta$' definition).
In shorhand notation
$$\frac{\mathrm{d}y}{\mathrm{d}x}:=\lim_{t\rightarrow 0}\frac{y(x+t)-y(x)}{t}.$$
$$$$
While the notation was inherited, it did no longer hold the original meaning.
And Riemann established his formalization of integration.
Based on these, people started to work on functions defined in real number system (real analysis). And in the meantime, the properties of real number were intensively explored (set theory, continuum, etc.).
Later the concept of limit was further extended to more general spaces, such as metric spaces(generalized distance), normed spaces(generalized length).
So many branches of analysis such as measure theory(it's a part of real analysis. I put it here simply because I feel it is so important.), functional analysis, differential equations emerged.
Hence, roughly speaking, changing from infinitesimal approach to limit approach can be considered as the line separating calculus and analysis.
Interestingly, in modern calculus textbooks, they in fact loosely use analysis approach while they remain name themself as Calculus. Is this because they do not discuss real number system, which is the very base for the rest. And they only loosely argue 'taking limit by $\Delta x\rightarrow 0$'? I really get confused here.
Updates:
Can I state that calculus is a study on real-valued functions with $\mathbb{R}^d$-valued argument?
So one can loosely conclude that
$$\text{infinitesimal and integral calculus} \subsetneq \text{real analysis}\subsetneq \text{analysis}.$$
Update again:
The question is much clear now.
If calculus is understood as art of calculation, there is no more confusions.
Thanks for all dedications on this topic!
Since most of answers pointed out the linchpin for the question, I hope it won't cause any misunderstanding if I do not accept any of them.
At the end, I hope this post will help others in future.
Cheers.
Best Answer
Remember, that the term "calculus" really stands for a method of calculation, especially one of several highly systematic methods of treating problems by a special system of ... notations. Think of vector and tensor calculus, calculus of variations, lambda calculus etc. Binding this term to integrals and derivatives is probably rather a pedagogical, than scientifical tradition.