Confusing about an integral's result

Questions
1

Hi,

I use following command to investigate profit rate in terms of stock at t (S[t]) and demand at t as weel as price at t (market-clearing). In the first three lines, I havent subsituite in a expression of S[t] in terms of demand[t]. And I have profit rate = Demand[t] p[t] - [Sigma] S[t]

However, when i substitute by

S[t] = S0 - \!\(\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(Demand[
     t] \[DifferentialD]t\)\)

(S0 is the initial stock), the result seems different from the expression for profit rate above. The first two parts of the result are the same as the expression above, but i don’t know where -t [Sigma] (-D0 + a p[t]) comes from. Could you help me explain this? (I guess it might be a math problem?)

Thank you

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2

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3

You are living dangerously by using the same variable t for both the integration variable and the endpoint of the interval of integration. It seems that an object of the form Integrate[f[t], {t, 0, t}] is treated in a funny way by Integrate:

D[Integrate[f[t], {t, 0, t}], t]
Integrate[Integrate[f[t], {t, 0, t}], {t, a, b}]

Is the following what you wanted?

Clear[s, demand];
demand[t_] = d0 - a p[t];
s[t_] = s0 - Integrate[demand[r], {r, 0, t}];
Integrate[p[u] demand[u], {u, 0, t}] - 
 Integrate[s[u] \[Sigma], {u, 0, t}]
profitRate[t_] = D[%, t]