The marginal revenue of a company is given by $MR = 80+20x+3x^2$, where $x$ is the number of units sold for a period. Find the total revenue function $R(x)$ if at $x=2$, $R(x)=240$.

Question

Accepted Answer

> [!TLDR]
> Revenue is the integral of marginal revenue; integrate $MR$ and fix the constant with $R(2)=240$.

> [!FORMULA]
> $$R(x)=\int MR\,dx,\qquad MR=\frac{dR}{dx}.$$

**Steps**

1. Integrate the marginal revenue:
   $$R(x)=\int(80+20x+3x^2)\,dx=80x+10x^2+x^3+C.$$
2. Apply $R(2)=240$:
   $$80(2)+10(2)^2+(2)^3+C=160+40+8+C=208+C=240.$$
3. Solve for $C$: $C=240-208=32.$
4. Therefore $R(x)=x^3+10x^2+80x+32.$ …

Q.The marginal revenue of a company is given by $MR = 80+20x+3x^2$ , where $x$ is the number of units sold for a period. Find the total revenue function $R(x)$ if at $x=2$ , $R(x)=240$ .

Concept understanding — Marginal Revenue Integration

Marginal Revenue Integration — From Intuition to Precision

The Intuition: Why MR Isn't Just Price

The Precise Statement

Why "Integration"?

Q.The marginal revenue of a company is given by MR=80+20x+3x2MR = 80+20x+3x^2MR=80+20x+3x2, where xxx is the number of units sold for a period. Find the total revenue function R(x)R(x)R(x) if at x=2x=2x=2, R(x)=240R(x)=240R(x)=240.

🎓Unlock everything free for 14 days

Q.The marginal revenue of a company is given by $MR = 80+20x+3x^2$ , where $x$ is the number of units sold for a period. Find the total revenue function $R(x)$ if at $x=2$ , $R(x)=240$ .

Unlock everything free for 14 days