Q.The marginal revenue function of a firm is $MR(x) = 50-4x$. Find the total revenue function $TR(x)$ and the demand function $p(x)$.

Total revenue is the integral of marginal revenue, with the constant of integration fixed by the fact that revenue is zero when no units are sold.

Step 1 — integrate the marginal revenue function:

Step 2 — fix the constant using $TR(0)=0$ (no units sold means no revenue):

So the total revenue function is:

Step 3 — find the demand function:

$p(x)=\frac{TR(x)}{x}=\frac{50x-2x^{2}}{x}=50-2x$ …

Fixing the constant of integration using $TR(0)=\text{Fixed Cost}$ by confusing this with the total-cost case above — revenue has no fixed component, so the c …