Let:
$C$ be a circle
$F$ be some physical quantity over $C$ (eg: force)
$f$ be a function on $C$
$F=\int_C f\ dA$
$P$ be any partition of the circle containing $n$ sub-intervals
$C_i$ be each subinterval
$F_i$ be that physical quantity over $C_i$
Then I need to prove:
$$F=\sum_{i=1}^{n} F_i$$
implies
$$\forall\ (i\ \leq n) \in \mathbb{N}, F_i=\int_{C_i} f\ dA$$
My try:
Since integral can be added and the physical quantity can be superposed: \begin{align} F &=\int_C f\\ = \sum_{i=1}^{n_1}F_i &=\sum_{i=1}^{n_1} \int_{C_i} f \ \text{(for partition $P_1$)}\\ = \sum_{i=1}^{n_2}F_i &=\sum_{i=1}^{n_2} \int_{C_i} f \ \text{(for partition $P_2$)}\\ = \sum_{i=1}^{n_3}F_i &=\sum_{i=1}^{n_3} \int_{C_i} f \ \text{(for partition $P_3$)}\\ &\vdots\\ \end{align}
Now how shall I proceed to show for any partition, for any subinterval, $$F_i=\int_{C_i} f$$?