From the marginal cost to the total cost.

[Francobati]

Hello. Can you help me figure out how to pass, integrating, by the marginal cost: $MC_{i}(q)_{i}=q_{i}+10$ to the total cost: $TC=\frac{1} {2}q_i^2+10q_{i}$?
$i=1,2$, are the two companies. $q_{i}$ is the quantity. What are the calculations?

 

[MarkFL]

Given a marginal cost $C_M$, a fixed cost $C_F$ and a quantity $q$, we are to assume (using the definition of marginal cost) for the total cost $C_T$:

$$\d{C_T}{q}=C_M$$

Now, if we integrate both sides w.r.t $q$, exchange the dummy variables of integration (and use a linear marginal cost function) and using the given boundaries, we obtain:

$$\int_{C_F}^{C_T}\,du=\int_0^q av+b\,dv$$

Applying the FTOC, there results:

$$C_T-C_F=\frac{a}{2}q^2+bq$$

Now, for this problem, it would appear there are no fixed costs ($C_F=0$), and we are given $(a,b)=(1,10)$, hence:

$$C_T=\frac{1}{2}q^2+10q$$