I don't know whether this question is suitable or not for this forum because it is not exactly rigorous mathematics.
We generally descibe geometrically absolute value of a definite integral as area under the curve(exception comes when the integrand changes its sign but even in that case also we can describe it geometrically and in this question it is irrelevant)
Now consider $$A = \int_{0}^{1} e^x \, dx$$
If we take our $x$ in meters or if we measure our coordinate axis in meters($m$) we see that $A$ has the unit of $m^2$ because it is area. Now obviously $dx$ has the unit of $m$ and $e^x$ is unitless. So RHS of the above equation has its unit in meters where as LHS has $m^2$. This perplexing inconsistency is haunting me. Can any one explain this paradox?
$e^x$ is not unitless. It is the distance from the $x$ axis measured in $m$ as well. Think of velocity, if an object moves with velocity $e^t$, this means that at time $t$ its velocity is $e^t$ something. Define the units of this something, do an integral and you'll know how far it went in these units.