Riemann Integral vs. Scalar Field Integral vs. Vector Field Integral (In single variable case)

Question

I am a bit confused about the relation between these three integrals. As my textbook doesn't seem to draw a very clear distinction between them. [this is somewhat related to this question Are regular/Riemann integrals a special case of a line integral?]

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Suppose there is a single-variable real-valued function $y=f(x)$

The Riemann integral is then: $$\int_a^bf(x)dx,\ \ \ \text{where$\int_a^b=-\int_b^a$}$$

The Vector Field line integral is then: $$\int_a^bf(x)\vec j\cdot \vec idx=0,\ \ \ \text{where$\int_a^b=-\int_b^a$}$$

The Scalar Field line integral is then: $$\int_a^bf(x)|dx|,\ \ \ \text{where$\int_a^b=\int_b^a$}$$

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The Scalar Field Integral has the same absolute value as the Riemann Integral, but different in the property $\int_a^b=-\int_b^a$

The Vector Field Integral has the same property $\int_a^b=-\int_b^a$ as the Riemann Integral, but different in the absolute value (in this particular case, the value of vector integral is zero).

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[Q1]: Are these three types of integral inherently different things? Or they can be treated as special cases of a generalized form

[Q2]: When parametrize a two variable scalar field line integral. $$\int_Cf(x,y)ds=\int_a^bf(t)\left\| {\,\vec r'\left( t \right)} \right\|dt$$ My textbook assumed that the parametrized integral is a Riemann integral, but why?

Answer 1

For Q1, even if an intro calculus book might not emphasize this, there are good reasons to think of what you're calling "scalar/vector field integrals" as different things. I recommend reading the beginning of Terry Tao's preprint on "differential forms and integration" which discusses that distinction.

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For Q2: The Riemann integral is a clarification in a different "direction". Both or neither of the above could be Riemann integrals. Basically, it's a Riemann integral if the underlying definitions with limits and sums is set up in a certain way, as opposed to some other way, like the (arguably more advanced) Lebesgue integral.