Applications of Definite Integrals (Area, Cost, Revenue)
This domain turns definite integrals into geometry and commerce: area between curves, and recovery of total revenue or total cost from marginal functions. Typical questions give MR or MC and ask for total or for the change over an interval.
Area under a curve
If f(x) ≥ 0 on [a, b], the area between the curve and the x-axis is ∫ₐᵇ f(x)dx. A region below the axis gives a negative value, so take the modulus or split the interval at the root. Between two curves the area is ∫ₐᵇ (upper − lower)dx; find intersection points first and split the interval where the curves cross.
Commerce: accumulating marginal functions
Total revenue from marginal revenue is TR = ∫₀ˣ MR(t)dt, because TR = 0 when nothing is sold. Total cost is TC = ∫₀ˣ MC(t)dt + fixed cost, since the constant of integration is exactly the fixed cost. The change in cost between outputs x₁ and x₂ is ∫ₓ₁ˣ² MC dx.
Worked example
MR = 10 − 2x rupees per unit. Then TR(x) = ∫₀ˣ (10 − 2t)dt = [10t − t²]₀ˣ = 10x − x². At x = 4, TR = 40 − 16 = ₹24. Substituting the lower limit 0 gives 0, matching the condition that revenue is zero at zero sales.
Traps
- Forgetting to add the fixed cost when integrating MC to obtain TC.
- Integrating without limits and then treating the constant as zero in a cost problem.
- Subtracting the lower curve from the upper the wrong way round.
- Ignoring an intersection point inside the interval, which splits the area into two integrals.