Integration and Applications
Integration and Applications tests antiderivatives, definite integrals and their business uses: recovering total functions from marginal functions, finding area under a curve, and computing consumer or producer surplus.
Core rules
- Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C.
- Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′(x) = f(x).
- If marginal cost is MC, total cost is C(x) = ∫MC dx + fixed cost. If marginal revenue is MR, total revenue is R(x) = ∫MR dx, with R(0) = 0.
- Area between a curve y = f(x), the x-axis and x = a, x = b is ∫ₐᵇ |f(x)| dx. Split the interval where the curve crosses the axis.
- Consumer surplus at (q₀,p₀) is ∫₀^{q₀} demand(q) dq − p₀q₀.
Worked example
A firm's marginal revenue is MR = 10 − 2x. Find total revenue and the demand function. R(x) = ∫(10 − 2x) dx = 10x − x² + C. Since R(0) = 0, C = 0. So R(x) = 10x − x². Demand price p = R/x = 10 − x. Check: differentiating R gives MR = 10 − 2x, as required.
Traps
Omitting + C in indefinite integrals; adding the fixed cost after rather than before evaluating a definite integral; reversing the limits and forgetting the sign change; integrating a marginal function but calling the result profit when it is revenue or cost; and treating area below the x-axis as positive without splitting the interval.