ECON1003 · Quantitative Methods In Economics
Integration
Integration undoes differentiation — to integrate a power, raise the index by one and divide by the new index, with the single exception of x⁻¹, which integrates to ln|x|. Every indefinite integral carries a + C; never drop it, because in differential-equation questions you solve for it later. Definite integrals turn this machinery into area, and area is how this course measures consumer and producer surplus. This chapter (Final-only, not on the midterm) drills the power / ln / eˣ rules, substitution and integration by parts (the by-parts formula is on the sheet), the definite integral read as area or as surplus, and differential equations — recovering total cost from marginal cost, or solving a separable growth law.
What this chapter covers
- 016.1 The integral as an anti-derivative and the constant + C
- 026.2 The power rule and the n = −1 (ln) exception
- 036.3 Substitution and integration by parts
- 046.4 The definite integral and area between curves
- 056.5 Consumer and producer surplus as areas
- 066.6 Differential equations: recovering TC from MC, separable growth
Worked example: recover total cost from marginal cost
- +1Set up. TC is the anti-derivative of MC: TC(Q) = ∫(6Q + 4) dQ.
- +1Integrate term by term. ∫6Q dQ = 3Q²; ∫4 dQ = 4Q. So TC = 3Q² + 4Q + C.
- +1Pin down C using the fixed cost. At Q = 0, TC = fixed cost = 50, so C = 50.
- +1State TC(Q). TC(Q) = 3Q² + 4Q + 50.
- +1(b) Evaluate at Q = 10. TC(10) = 3(100) + 4(10) + 50 = 300 + 40 + 50 = $390.
Key terms
- Anti-derivative
- A function g whose derivative is f: g′(x) = f(x). Because the derivative of a constant is zero, anti-derivatives are only pinned down up to an additive constant — hence the constant of integration C.
- Constant of integration (C)
- The + C on every indefinite integral. It matters: in a differential-equation question (recovering total cost from marginal cost, or a growth law) you solve for C using a known condition such as TC(0) = fixed cost.
- Definite integral
- An integral evaluated between two limits, ∫ᵃᵇ f(x)dx, which has no + C and gives a number — the signed area under the curve between a and b. It is how the course measures surplus.
- Consumer / producer surplus
- Areas computed by definite integration: consumer surplus is the area between the demand curve and the price; producer surplus is the area between the price and the supply curve, up to the traded quantity.
- Separable differential equation
- An ODE you solve by separating the variables onto opposite sides and integrating each, then using a known condition to find the constant. The course uses it for growth laws and for recovering a total from a marginal.
Integration FAQ
Why must I never drop the + C?
Because the indefinite integral is only defined up to a constant (any constant differentiates to zero). In differential-equation questions — recovering total cost from marginal cost, or solving a growth law — you use a known condition (such as TC(0) = fixed cost) to solve for C. Drop it and you cannot complete the problem. Definite integrals are the only place C disappears.
What's special about integrating x⁻¹?
The power rule says raise the index by one and divide by the new index — but at n = −1 that would divide by zero. So this single case becomes the natural log: ∫(1/x)dx = ln|x| + C. Every other power follows the standard rule; only x⁻¹ is the exception, and it is a favourite exam check.
How does integration measure surplus?
A definite integral is an area. Consumer surplus is the area between the demand curve and the market price (what buyers were willing to pay above what they paid); producer surplus is the area between the price and the supply curve. You set the integral between the relevant quantity limits and evaluate. Sketching the area first prevents sign and limit errors.
Exam move
Integration is Final-only, so schedule it for the back half of revision but do not skip it — it carries the surplus and differential-equation marks. Make the + C automatic and remember the one exception, x⁻¹ → ln|x|. For substitution and by-parts, practise spotting which technique a integrand wants (by-parts is on the sheet; the rest you memorise). For the economic applications, always sketch the area before integrating surplus, and treat 'recover TC from MC' as a fixed routine: integrate MC, then pin down C from the fixed cost. Show the substitution and the limits; method marks are real.
