Mathematics of producer surplus wrong?

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The formula for producer surplus is debated, with one perspective suggesting it should include fixed costs (FC) in the equation, resulting in P*Q - (∫M(Q)dQ + FC). This argument highlights that marginal costs, derived from total costs, only account for variable costs when integrated, potentially overlooking fixed costs that impact overall profitability. Additionally, there is a discussion on demand curves, questioning whether they accurately represent the quantity the market is willing to buy. This raises important considerations about how producer surplus is calculated and the implications for understanding market dynamics.
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The formula for producer surplus is:

Income - expenses = P*Q - ∫M(Q)dQ

However, shouldn't it be P*Q - (∫M(Q)dQ + FC), with FC= fixed costs?

I mean, the marginal costs are just the derivative of total costs, and thus integrating them is the same as just integrating the variable costs, ignoring the fixed costs of production.
 
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And concerning demand-curves:

Do they represent the quantity the market is willing to buy?
 
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