Question

how do you find variable cost using marginal cost?

Answer 1

I guess an Integral of a variable cost is its Marginal cost

Answer 2

Dear cvilla1

Remember that if \[CT(q)=CV(q)+F\] is the total cost, \[CV(q)\] is the variable cost, \[F\] is the fixed cost and \[q \ge0\] is the quantity then the marginal cost is define as:
Discrete case: \[\frac{ \Delta CT(q) }{ \Delta q } = \frac{ \Delta CV(q) }{ \Delta q } = \frac{ CV(q + \Delta q) -CV(q)}{ \Delta q }\]
If \[\Delta q = 1\] then \[\frac{ \Delta CT(q) }{ \Delta q } = CV (q+1) - CV(q)\]
For the discrete case
\[CV(q) = \sum_{q=0}^{q-1} [CV(q+1) - CV(q)] \] where \[CV(0) = 0\]
Continuous case: \[\lim_{\Delta q \rightarrow 0} \frac{ \Delta CT(q) }{ \Delta q } = \lim_{\Delta q \rightarrow 0} \frac{CT(q + \Delta q) - CT(q)}{ \Delta q } = \frac{ dCT(q) }{ dq } \]
For the continuous case by the Fundamental Theorem of Calculus:
\[CT(q) = \int\limits_{0}^{q} \frac{ dCT(q) }{ dq } dq\]
Therefore
\[CV(q) = CT(q) - F = \left( \int\limits_{0}^{q}\frac{ dCT(q) }{ dq }dq \right) - F\]
Best wishes

luifrancgom