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

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?]

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: $$\int_a^bf(x)|dx|,\ \ \ \text{where\int_a^b=\int_b^a}$$

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).

[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?