Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)? I know that (co)ends (i.e. universal wedges) follow Fubini-like relation, i.e.
$$ \int_{\langle c,d\rangle} F(c,d,c,d) \cong \int_c\int_d F(c,c,d,d) \cong \int_d\int_c F(c,c,d,d) $$
where we regard $F$ as both a functor $(C\times D)^{op}\times (C\times D)\to X$ and a functor $C^{op}\times C \times D^{op} \times D \to X$. 
Is there any other reason we use this notation? E.g. is their formulation as a limit in some way analogous to Riemann sums, do they have some kind of linearity relation, etc?
 A: Since coends commute with colimits, in particular coproducts, we have some kind of "linearity"
$$\int^i \bigl(X(i,i) \oplus Y(i,i)\bigr) \cong \int^i X(i,i) \oplus  \int^i Y(i,i).$$
Every functor $F : C \to D$ into a cocomplete category $D$ may be decomposed as
$$F \cong \int^{x \in C} F(x) \otimes \hom(x,-),$$
this is basically a reformulation of the Yoneda Lemma. Here, $\otimes$ denotes the copower $D \times \mathsf{Set} \to D$.
This looks very similar to the equation (in fact, this is how I remember this)
$$f(y) = \int_{x \in \Omega} f(x) \cdot d \delta_y(x),$$
where $\delta_y$ denotes the dirac measure concentrated at $y \in \Omega$ and $f : \Omega \to \mathbb{R}$ is a measurable function, as well as to the equation
$$\mu(A) = \int_{x \in \Omega} d \mu(x) \cdot \chi_A(x)$$
for measures $\mu$ on $\Omega$.
However, the analogy between coends and integrals, in particular between coproducts and sums doesn't go too far: The sum of real numbers has no analogue of a universal property. There are no analogues of coproduct inclusions. In category theory there are no nontrivial objects which are invertible with respect to coproducts. And sums in analysis are not really dual to products. Finally notice that Fubini's theorem in category theory is a formal consequence of the definitions, whereas Fubini's theorem in measure theory is not.
A: It's perhaps not great notation, but some of the most common types of coends arising in practice (namely, weighted colimits) can be thought of roughly as "categorified weighted sums". 
In enriched category theory (over a complete, cocomplete symmetric monoidal closed category $V$), a weight consists of a small $V$-category $J$ and a $V$-functor $W: J^{op} \to V$, and the weighted colimit of a functor $F: J \to C$ over the weight $W$ is an object $c$ together with a $V$-natural isomorphism 
$$C(c, d) \cong V^{J^{op}}(W-, V^J(F-, d))$$ 
(where as usual in this subject, $\hom_C(d, d)$ is abbreviated to $C(c, d)$). The typical way this colimit is presented, for a $V$-cocomplete category $C$, is by means of (as always, $V$-enriched) coends and tensors (where we denote tensors with a dot symbol $\cdot$):
$$c = \int^{j: J} W j \cdot F j.$$ 
Let's consider the most vanilla case possible, where $V$ is just $\text{Set}$ and $J$ is small discrete (in other words, a set). A weight in this case just assigns a set $W(j)$ to each element $j \in J$; the $W(j)$ might be finite or infinite, but no matter. Then this coend formula just amounts to a coproduct 
$$c = \sum_{j \in J} W(j) \cdot F(j)$$ 
where $W(j) \cdot F(j)$ means we are taking a coproduct of a bunch of copies of $F(j)$, one for each element $x \in W(j)$. In other words, the weighted colimit is a categorified weighted sum of $F$, somewhat analogous to weighted sums in analysis (e.g., integration of a function $f$ against a measure $d\mu = w(x) dx$ as in Lebesgue-Stieltjes integration). 
The more general weighted colimits are not coproducts of course, but they are related. Again, to take the vanilla case $V = \text{Set}$, the coend formula amounts to a coequalizer appearing in a diagram:  
$$\sum_{j, k \in J} W(j) \cdot J(k, j) \cdot F(k) \rightrightarrows \sum_j W(j) \cdot F(j) \to \int^j W(j) \cdot F(j)$$ 
where the two parallel arrows are induced by canonical maps $W(j) \cdot J(k, j) \to W(k)$ and $J(k, j) \cdot F(k) \to F(j)$, which in turn owe their presence to the functorial structure of $W$ and $F$. 
