$\int^a_b f(x) dx$ is the sum of the areas of infinitesimal strips under the graph $f(x)$. In that sense, the integral is not a continuous sum of $f(x)$ between $f(a)$ and $f(b)$, but rather a continuous sum of $f(x)dx$ between those values, where $f(x)dx$ is the area of each strip. The result of the sum therefore is to give us the total area under the graph between $x=a$ and $x=b$. Why, then, do you so many physics textbooks treat such an integral as the continuous case of $\sum_i f(x_i)$, when this is not a sum of the area?
An example from Griffiths's Quantum Mechanics:
We have a solution to the Schrodinger equation for a particle in an infinite potential well given by:
$$\Psi_n=c_n\sin(\frac{n\pi}{a}x)e^{-i\frac{n^2\pi^2\hbar}{2ma^2}t}\ \ \forall \ \ n \in \mathbb{N}$$
Therefore the general solution is given by:
$$\Psi=\sum_{n=1}^\infty c_n\sin(\frac{n\pi}{a}x)e^{-i\frac{n^2\pi^2\hbar}{2ma^2}t}$$
For the case of a free particle (no potential function) a solution is given by:
$$\Psi_k = c_ke^{i(kx-\frac{\hbar k^2}{2m}t)} \ \ \forall \ \ k \in \mathbb{R}$$
Therefore the general solution is given by:
$$\Psi = \int_{-\infty}^\infty c(k)e^{i(kx-\frac{\hbar k^2}{2m}t)}dk$$
How do we justify the presence of the $dk$?