oh, no no no. its absolutely not like that. e is just a constant, just like any other constants, eg pi. etc
so when u integrate a constant function, ur just basically finding the area of the rectangle the function makes between the two bounds of the definite integral. for example:
[math]\int_1^{2}3dx=[3x]_1^{2}=6-3=3[/math]
which is the same thing as finding the area of the rectagle with height 3 and length (2-1)=1.
while e is not a function of anything. e^x is a function. its quite a nice function as the value of its derivative at a point is equal to the value of the function at that point.
i am not really good at explaining stuff. maybe someone will help you better