Not an "opinionated" book per se, but Tom Apostol's Calculus follows up the chronological order of the concepts of the calculus.

Hence, it starts with an example of Archimedes exhaustion method and after defines the integrals before limits as:

Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t $ denote arbitrary step functions defined on [a, b] such that

$s(x) \leq f(x) \leq t(x)\hspace{3cm}(1)$

for every $x$ in $[a, b]$. If there is one and only one number $I$ such that:

$\int_{a}^{b}s(x)dx \leq I \leq \int_{a}^{b}t(x)dx$

for every pair of step functions $s$ and $t$ satisfying (1), then this number $I$ is called the integral of $f$ from $a$ to $b$, and is denoted by the symbol $\int_a^bf(x)dx$ or $\int_a^bf$.

Later, after the integral is defined and proved solely based on the integration of step functions, the $\epsilon - \delta$ is introduced.

It is an interesting approach, but Apostol never tries to convince us that it is a superior one, so it is not really an "opinionated" book. However, if we consider that writing a two volume textbook is not easy, it does contain the indirect opinion that, at least, Apostol liked this approach better. Also, in his words:

In this book the subject is introduced in an informal way, and ample use is made of geometric intuition whenever it is convenient to so so."

Which are now commonplace in many calculus textbooks.

Not an "opinionated" book per se, but Tom Apostol's Calculus follows up the chronological order of the concepts of the calculus.

Hence, it starts with an example of Archimedes exhaustion method and after defines the integrals before limits as:

Let $f$ be a function defined and bounded on $[a, b]$. Let $s$ and $t $ denote arbitrary step functions defined on [a, b] such that

$s(x) \leq f(x) \leq t(x)\hspace{3cm}(1)$

for every $x$ in $[a, b]$. If there is one and only one number $I$ such that:

$\int_{a}^{b}s(x)dx \leq I \leq \int_{a}^{b}t(x)dx$

for every pair of step functions $s$ and $t$ satisfying (1), then this number $I$ is called the integral of $f$ from $a$ to $b$, and is denoted by the symbol $\int_a^bf(x)dx$ or $\int_a^bf$.

Later, after the integral is defined and proved solely based on the integration of step functions, the $\epsilon - \delta$ is introduced.

It is an interesting approach, but Apostol never tries to convince us that it is a superior one, so it is not really an "opinionated" book. However, if we consider that writing a two volume textbook is not easy, it does contain the indirect opinion that, at least, Apostol liked this approach better. Also, in his words:

In this book the subject is introduced in an informal way, and ample use is made of geometric intuition whenever it is convenient to so so.

Which are now commonplace in many calculus textbooks.