# Integral Calculus

Business I · Mathematics I · https://tryals.app/learn/business-i/integral-calculus

## Adding Up the Margin

If the derivative slices totals into margins, the **integral** reassembles margins into totals. The two operations undo each other, which is the **Fundamental Theorem of Calculus**:

$$\int_a^b f(x)\,dx = F(b) - F(a), \qquad F' = f$$

**Antiderivatives**: $F$ is an antiderivative of $f$ if $F' = f$. The power rule works in reverse for integration:

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

**The constant of integration**: The $+C$ matters in economics. Knowing marginal cost pins total cost only **up to fixed costs**; the constant is the part the margin cannot see.

> **Common pitfall:** Dropping the constant of integration. Marginal cost knows nothing about fixed costs, integrating it recovers total cost only **up to** fixed costs, and forgetting that constant understates every cost estimate by exactly the rent.

## Economic Areas

The definite integral measures accumulated totals, and in business, those areas represent money. Integration turns rates and margins into totals:

| Integrate… | Over… | You get |
|---|---|---|
| Marginal cost | $0$ to $q$ | Total variable cost |
| Sales rate | A month | Total units sold |
| Marginal profit | Two outputs | Profit **change** |

**Consumer surplus**: The area between the demand curve and the market price $p^*$. Everyone pays $p^*$, but early buyers pay less than their willingness to pay:

$$CS = \int_0^{q^*} \big(D(q) - p^*\big)\,dq$$

For linear demand, this equals the triangle area: $CS = \tfrac{1}{2}\,q^*\,(p_{max} - p^*)$. **Producer surplus** mirrors this below the price line and above supply, measuring total trade value.

## Practice questions

4 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Derivative or integral? Sort each business calculation by the tool it needs.

**Answer:**

- Differentiate: From total cost, find the cost of the next unit, From revenue $R(q)$, find marginal revenue
- Integrate: From a daily sales rate, find monthly volume, From marginal profit, find profit change between two outputs, From a cash-flow rate, find cash accumulated this quarter

**Why:** Totals → margins: differentiate. Rates/margins → totals: integrate. Every finance spreadsheet is quietly doing one or the other.

Page: https://tryals.app/practice/business-i/integral-calculus/derivative-or-integral-sort-each-business-calculation-by-the-tool-it

### 2. Match each integral to its business meaning.

**Answer:**

- $\int_0^q MC(x)\,dx$ → Total variable cost of producing $q$
- $\int_0^{q^*}(D - p^*)\,dq$ → Consumer surplus
- $\int_{q_1}^{q_2} M\pi(q)\,dq$ → Extra profit from raising output
- $\int_0^{30} s(t)\,dt$ → Units sold over a 30-day month

**Why:** Same symbol, four dashboards: cost, welfare, profit deltas, and volumes, all areas under a margin or rate.

Page: https://tryals.app/practice/business-i/integral-calculus/match-each-integral-to-its-business-meaning

### 3. A CFO knows marginal cost perfectly but says total cost is still undetermined. Why is she right?

A. Integration recovers totals only up to a constant
B. Marginal cost cannot be integrated
C. She is wrong: margins determine totals exactly
D. Total cost is always lower than integrated marginal cost

**Answer:** A. Integration recovers totals only up to a constant

**Why:** Antiderivatives come with $+C$: the margin sees only how cost *changes* per unit, never the rent, licenses, and salaries paid at $q = 0$. The constant of integration is the accountant’s fixed cost.

Page: https://tryals.app/practice/business-i/integral-calculus/a-cfo-knows-marginal-cost-perfectly-but-says-total-cost-is-still

### 4. Two firms have identical marginal cost curves and different total costs at every output. How is that possible?

A. The firms produce different quantities, which changes total cost
B. It is impossible; identical marginal costs force identical totals
C. They differ by a constant of integration, the fixed costs
D. One firm must be measuring marginal cost over a different interval

**Answer:** C. They differ by a constant of integration, the fixed costs

**Why:** The derivative annihilates constants, so marginal cost is silent about fixed costs, the integral recovers the shape and needs one known point to place it. This is why a firm can price on margin and still lose money, and why break-even analysis needs the constant back.

Page: https://tryals.app/practice/business-i/integral-calculus/two-firms-have-identical-marginal-cost-curves-and-different-total