Expected profit given uniformly distributed marginal costs in a model of Betrand Competition

Question

I am struggling with the following problem: Suppose there are 3 firms, A B and C, in the market which are competing by setting their prices. Firm A has marginal costs of $a$ which are between 0 and 1. Firm B has marginal costs $b$ which are between 0 and 1. Firm C has marginal costs of $c$, where $c$ is uniformly distributed in the interval from 0 to 1. Further, we know that $a$ is always smaller than $b$. We assume that firm C and firm B have additional costs of $\tau$ for selling their product. That means, total marignal costs of C are $c+\tau$ and for B $b+\tau$. Demand is given by $$D=M-p$$ For $$a < c+\tau < b+\tau$$ the whole market is served by A and price is given by $$p=c+\tau$$ What is the expected profit of A for $a<c+\tau<b+\tau$? I calculated: $$pi_A(a < c+\tau < b+ \tau)=\int_{a-\tau}^{b} (c+\tau-a)(M-(c+\tau))\ dc_R$$ or is it : $$pi_A(a < c+\tau < b+ \tau)=\int_{a}^{b+\tau} (c+\tau-a)(M-(c+\tau))\ dc_R$$ I am not sure which one it is, since I need the expected value of c and not of $c+\tau$. Further, I do not really understand, why the results of the integrals are not the same. I thought, since $c$ is uniformly distributed, it does not matter whether the integral is from $a-\tau$ to $s$ or from $a$ to $s-\tau$ because in the end, the area is the same. But, maple told me otherwise: Equation (1) - equation (2) is not zero. I appreciate any help!! Thanks.