ECON5005 · Quantitative Tools for Economics
Integration
Integration is the closing calculus topic of ECON5005 Quantitative Tools for Economics at the University of Sydney, and it runs differentiation in reverse. This chapter builds the antiderivative and its rules (power rule, the 1/x exception, exponentials, linearity), adds substitution and integration by parts, and evaluates definite integrals as signed area via the Fundamental Theorem. The economic pay-off is recovering a total from a marginal — total revenue from MR and total cost from MC — pinning the constant with a boundary condition, and reading consumer surplus off as an area.
What this chapter covers
- 01Find indefinite integrals with the power rule, and handle the n = -1 case as ln|x| + C
- 02Integrate exponentials and use linearity to split sums and pull out constants
- 03Reverse the chain rule with substitution: set u = the inner function
- 04Reverse the product rule with integration by parts: choose u to simplify on differentiation
- 05Evaluate definite integrals as F(b) - F(a) and read them as signed area
- 06Recover TR from MR and TC from MC by integrating, then pin C with a boundary condition
- 07Use TR(0) = 0 for revenue and TC(0) = fixed cost for cost
- 08Compute consumer surplus as the definite integral between demand and the price line
Recover TR and TC from marginals, then optimise
- +1Integrate MR: TR = INT(20 - 4Q)dQ = 20Q - 2Q^2 + C. Impose TR(0) = 0, so C = 0 and TR(Q) = 20Q - 2Q^2.
- +1Integrate MC: TC = INT(2 + 2Q)dQ = 2Q + Q^2 + C. Impose TC(0) = 5 (the fixed cost), so C = 5 and TC(Q) = Q^2 + 2Q + 5.
- +1Assemble profit: pi(Q) = TR - TC = (20Q - 2Q^2) - (Q^2 + 2Q + 5) = -3Q^2 + 18Q - 5.
- +1First-order condition MR = MC: 20 - 4Q = 2 + 2Q, so 18 = 6Q and Q* = 3.
- +1Second-order condition: pi'' = -6 < 0 (equivalently MR' = -4 < MC' = +2), so Q* = 3 is a maximum.
- +1Price from inverse demand P = TR/Q = 20 - 2Q, so P* = 20 - 2(3) = 14.
- +1Maximum profit: pi(3) = -3(9) + 54 - 5 = 22 (check: TR(3) = 42, TC(3) = 20, pi = 22).
- +1Consumer surplus: CS = INT_0^3 [(20 - 2Q) - 14] dQ = INT_0^3 (6 - 2Q) dQ = [6Q - Q^2]_0^3 = 18 - 9 = 9.
Key terms
- Indefinite integral (antiderivative)
- The family of functions whose derivative is f: INT f(x) dx = F(x) + C, where F'(x) = f(x). The arbitrary constant C appears because the derivative of any constant is zero.
- Constant of integration (C)
- The undetermined constant on every indefinite integral. In economics it is pinned by a boundary condition — zero for revenue via TR(0) = 0, and the fixed cost for cost via TC(0).
- Power rule for integration
- INT x^n dx = x^(n+1)/(n+1) + C for n =/= -1. The single exception is INT x^(-1) dx = ln|x| + C.
- Substitution
- The reverse chain rule: for INT f(g(x)) g'(x) dx, set u = g(x) so du = g'(x) dx and the integral becomes INT f(u) du. Convert back to x (or change the limits) at the end.
- Integration by parts
- The reverse product rule: INT u dv = uv - INT v du. Choose u to be the factor that simplifies when differentiated (a polynomial or a logarithm) and dv to be the factor you can integrate.
- Definite integral
- INT_a^b f(x) dx = F(b) - F(a) by the Fundamental Theorem of Calculus. It has no +C (the constant cancels) and equals the signed area between the curve and the x-axis.
- Recovering totals from marginals
- Because MR = TR' and MC = TC', integrating a marginal gives the total: TR = INT MR dQ and TC = INT MC dQ, each completed by its boundary condition.
- Consumer surplus
- The area between the demand curve and the price line, computed as a definite integral of (demand price - actual price) over the quantity traded.
Integration FAQ
When should I use substitution versus integration by parts?
Use substitution when the integrand contains an inner function and (a constant multiple of) its own derivative — for example INT 6x(3x^2+1)^4 dx, where 6x is the derivative of the inside 3x^2+1; set u equal to the inner function. Use integration by parts for a product of two different families, such as a polynomial times an exponential or a logarithm (INT x e^x dx): apply INT u dv = uv - INT v du, choosing u to be the factor that gets simpler when differentiated. A quick test: if you can spot 'a function and its derivative', substitute; if you see 'a product of unlike things', use parts.
How do I recover total revenue or total cost from a marginal function?
Integrate the marginal and then pin the constant of integration with a boundary condition. Total revenue is TR = INT MR dQ, and since a firm selling nothing earns nothing, TR(0) = 0 fixes the constant at zero. Total cost is TC = INT MC dQ, and TC(0) equals the fixed cost — usually not zero — so the constant equals that fixed cost. Skipping the boundary condition, or defaulting the cost constant to zero, changes profit and everything computed from it, so it is treated as a marked step.
Can AI help me with integration in ECON5005?
Yes, as a study aid. Sia, the AskSia AI tutor, can explain the method step by step: which technique fits an integrand, how substitution and by parts work, how to evaluate a definite integral, and how to recover TR or TC from a marginal and pin the constant — using practice problems with your own numbers so you learn the reasoning. It does not sit your quizzes, mid-semester or final exam or hand you assessment answers, and it cannot promise a grade; treat it as a tutor that helps you work each step yourself, and confirm the exact assessment rules on Canvas.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Integration rewards drilling the pattern-recognition, not memorising a table. Practise sorting an integrand into power rule, substitution (a function and its derivative both present) or by parts (a product across families), and write the +C every time until it is automatic. For economic questions, rehearse the full chain end to end: integrate the marginal, pin the constant with the boundary condition (TR(0) = 0 for revenue, TC(0) = fixed cost for cost), assemble profit, then apply MR = MC and the second-order condition before reporting price, profit and any surplus area. Because integration is a Week-12 topic it appears on the 2-hour in-person final rather than the mid-semester test; practise past-style marginal-to-total-then-optimise questions under time at roughly 2 minutes per mark. Confirm the exact assessment weights, and the open/closed-book and calculator or formula-sheet policy, on your Canvas page.
