Integrating hat functions

Question

I am trying to integrate a hat function for a project that I am doing and have found a method to do so but I find it sloppy. Currently I have the basis function

\[Psi][z_] := z - Subscript[Z, i]/ \[CapitalDelta]z + 1;

which I am trying to integrate from $z_{i-1}$ to $z_{i+1}$. I break the basis function up into two pieces and integrate the left side from $z_{i-1}$ to $z_{i}$ and then the right side from $z_i$ to $z_{i+1}$. My first question is, is there a way to integrate piecewise functions? The second question I have is, is there a way to set global assumptions like $z_{i-1} < z_i < z_{i+1}$, $z_i - z_{i-1} = \Delta z$ , etc?

Edit: This is the piece wise function taken directly from my code I am trying to integrate

\[Psi][z_, c_] :=  Piecewise[{{(z - c)/\[CapitalDelta]z + 1, 
 z <= c}, {-(z - c)/\[CapitalDelta]z + 1, z > c}}];

where $c$ is the center of the hat function. Here is my attempt to integrate the piece wise function

 FullSimplify[
 Integrate[\[Psi][z, Subscript[Z, i]], {z, Subscript[Z, i - 1], 
   Subscript[Z, i + 1]}], 
 Assumptions -> {-(Subscript[Z, i + 1] - Subscript[Z, 
       i ]) == -\[CapitalDelta]z, 
   Subscript[Z, i + 1] - Subscript[Z, 
     i ] == \[CapitalDelta]z, -(Subscript[Z, i] - Subscript[Z, 
       i - 1 ]) == -\[CapitalDelta]z, 
   Subscript[Z, i] - Subscript[Z, i - 1 ] == \[CapitalDelta]z}]

I do not get a usable answer. Am I doing something wrong (ie can one integrate a piece wise function)?

Answer 1

Short answer: Mathematica has no problem integrating piecewise or hat functions.

Your notation seems to me to be needlessly complex. Why bother to define $Z_i$ when it's just $Z_0+i\Delta z$? Isn't your $\psi$ just 1-Abs[z-c]/Δz? However, I'll try to adhere to the spirit of your notation. Here's some code that works for me:

Zdefs = {Subscript[Z, i] - Subscript[Z, i - 1] == Δz, 
   Subscript[Z, i + 1] - Subscript[Z, i] == Δz};
Integrate[ψ[z, Subscript[Z, i]],
   {z, Subscript[Z, i - 1], Subscript[Z, i + 1]} /. 
  Solve[Zdefs, {Subscript[Z, i + 1], Subscript[Z, i - 1]}][[1]], 
 Assumptions -> Δz > 0 && 
   Subscript[Z, i] ∈ Reals]

Answer 2

Your integral would have worked too, if you had added assumptions that all variables are real:

(* New assumptions so Integrate can do its work *)
Assuming[{z \[Element] Reals, 
  c \[Element] Reals, \[CapitalDelta]z \[Element] Reals, 
  Subscript[Z, i - 1] \[Element] Reals, 
  Subscript[Z, i + 1] \[Element] Reals},

 (* The rest is just copied from the question *)
FullSimplify[
 Integrate[\[Psi][z, Subscript[Z, i]], {z, Subscript[Z, i - 1], 
   Subscript[Z, i + 1]}], 
 Assumptions -> {-(Subscript[Z, i + 1] - Subscript[Z, 
       i ]) == -\[CapitalDelta]z, 
   Subscript[Z, i + 1] - Subscript[Z, 
     i ] == \[CapitalDelta]z, -(Subscript[Z, i] - Subscript[Z, 
       i - 1 ]) == -\[CapitalDelta]z, 
   Subscript[Z, i] - Subscript[Z, i - 1 ] == \[CapitalDelta]z}]]

Result:

$\begin{array}{cc} \{ & \begin{array}{cc} \frac{3 \left(Z_i-Z_{i+1}\right){}^2}{\text{$\Delta $z}} & Z_i>Z_{i+1} \\ \frac{\left(Z_i-Z_{i+1}\right){}^2}{\text{$\Delta $z}} & Z_i<Z_{i+1} \\ \end{array} \\ \end{array}$