In many first-year calculus courses, one should not teach Riemann integration nor Lebesgue integration nor any other theory of integration. An integral is an area under a curve, but also any of various other sums of infinitely many infinitely small quantities. The expression $\int_a^b f(x,y)\,dy$, the "$dx$" serves not only to identify which variable one is integrating with respect to, but also what the units are (e.g. of $f(x,y)$ is in meters per second and $dx$ is in seconds, then ..... etc.....).
I teach what I call the "boundary rule" on the first day of a calculus course: $$\left[\text{size of boundary}\right] \times \left[\text{rate of motion of boundary}\right]$$ $$= \left[\text{rate of change of size of bounded region}\right].$$ With concrete examples, of course. That's half of the Fundamental Theorem, and the other half can be given later, after one begins to talk about integrals. The product rule is a corollary: think of a rectangle with two moving sides. Proving that the area of a circle is $\pi r^2$ is another corollary. So are some other things that are less important but are good exercises.