Limits of integration of a multivariable function?

Question

My book is filled with problems such as $\int_{1}^{x} f(x, y) dy$ . My question is if there is a good reason why the top limit of integration is an $x$. When we integrate the function with respect to $y$, we are holding $x$ constant, so for example we could say $x=5$ . That means that in this definite integral, we are finding the area beneath a curve in the $yz$ plane at $x=5$. However, we are limited to only finding the area between $1$ and $5$. If we pick $x=6$, we are finding the area of a (potentially) different curve in the $yz$ plane at $x=6$, but we are limited to only finding the area between $1$ and $6$. So why would we restrict our definite integral like that, when we could just make the top limit of integration $t$ and then on any given curve in the $yz$ plane at $x=a$ we can find the area between $1$ and $t$, a variable, rather than between $1$ and $a$, a constant.