Quantitative Aptitude · Differential and Integral Calculus

Definite Integrals and Their Applications for CA Foundation

Updated 1 October 2026 · Fact-checked

A definite integral ∫ from a to b of f(x) dx gives the net accumulated value of f(x) between x = a and x = b. Find the antiderivative F(x), then compute F(b) − F(a). It is used for area under a curve, total cost or revenue from marginal functions, and consumer and producer surplus.

Understand Definite Integrals and Their Applications

Integration is the reverse of differentiation. An indefinite integral gives a family of functions with a constant C. A definite integral has limits a and b, so the constant cancels and you get a single number.

The number has a meaning. If f(x) is positive, ∫ from a to b of f(x) dx is the area between the curve, the x-axis and the lines x = a and x = b. If f(x) is a rate (like marginal cost), the integral is the total change in the quantity between a and b.

The link between the two ideas is the Fundamental Theorem of Calculus: if F′(x) = f(x), then ∫ from a to b of f(x) dx = F(b) − F(a). You do not need to add limits one by one. You only need the standard integration formulas.

In business, marginal cost MC is the derivative of total cost C. So the extra cost of going from output a to output b is the integral of MC from a to b. To get the total cost itself, integrate MC and use the fixed cost to find the constant. The same idea works for revenue and marginal revenue.

Consumer surplus is the gain buyers get because they pay the market price, not the higher price they were willing to pay. Producer surplus is the gain sellers get because they receive the market price, not the lower price they were willing to accept. Both are areas between a curve and the horizontal line at the equilibrium price.

Key formulas to remember

Fundamental theorem: ∫ from a to b of f(x) dx = F(b) − F(a), where F′(x) = f(x); Write the antiderivative in brackets with limits, then subtract lower from upper. No constant C is needed.
Power rule: ∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1), for n ≠ −1; For n = −1, ∫ (1/x) dx = log x (natural log, x > 0).
Exponential rule: ∫ eˣ dx = eˣ; Also ∫ eᵃˣ dx = eᵃˣ ÷ a for a ≠ 0.
Same limits: ∫ from a to a of f(x) dx = 0; No width means no area.
Reversing limits: ∫ from a to b of f(x) dx = − ∫ from b to a of f(x) dx; Swapping limits changes the sign.
Constant multiple and sum: ∫ k·f(x) dx = k·∫ f(x) dx; ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx; Applies to the same limits.
Splitting the interval: ∫ from a to b of f(x) dx = ∫ from a to c of f(x) dx + ∫ from c to b of f(x) dx; Useful when the curve changes sign or form at c.
Change of variable name: ∫ from a to b of f(x) dx = ∫ from a to b of f(t) dt; The letter used does not matter.
King property: ∫ from a to b of f(x) dx = ∫ from a to b of f(a + b − x) dx; Used to simplify some integrals, such as when the result gives 2I = something simple.
Even and odd functions: ∫ from −a to a of f(x) dx = 2 ∫ from 0 to a of f(x) dx if f is even; = 0 if f is odd; Even: f(−x) = f(x). Odd: f(−x) = −f(x).
Area under a curve: Area = ∫ from a to b of y dx, when y ≥ 0 on [a, b]; If y is below the axis, the integral is negative; take the absolute value for area.
Total cost from marginal cost: C(x) = ∫ MC dx + k, where k = fixed cost (since C(0) = k); Extra cost from x = a to x = b is ∫ from a to b of MC dx.
Total revenue from marginal revenue: R(x) = ∫ MR dx, with R(0) = 0; No units sold means no revenue, so the constant is 0.
Consumer surplus: CS = ∫ from 0 to x₀ of D(x) dx − p₀·x₀; D(x) is the demand price at quantity x. (x₀, p₀) is the equilibrium point.
Producer surplus: PS = p₀·x₀ − ∫ from 0 to x₀ of S(x) dx; S(x) is the supply price at quantity x.

How to solve Definite Integrals and Their Applications questions

Use this method for any definite integral or application question.

1Identify the type: pure integral, area, cost or revenue from a marginal function, or surplus.
2Write the integrand clearly. Expand brackets or split fractions so each term has a standard form.
3Integrate term by term using the standard formulas. Do not add C.
4Put the antiderivative in square brackets with the limits.
5Substitute the upper limit, then the lower limit, and subtract: F(b) − F(a).
6For cost or revenue, find the constant from the given condition (fixed cost, or zero revenue at zero output) before substituting.
7For surplus, first find the equilibrium point by setting demand equal to supply. Then apply the surplus formula.
8Check sign and units. Area and surplus must be positive. Write the answer in rupees or units if asked.

Quickest way: Option-check and shortcut approach

When to use it: Use this in the MCQ paper when time is short and the integrand is a simple polynomial or exponential.

Check for shortcuts first: same limits give 0; an odd function over −a to a gives 0; an even function gives twice the half-range value.
For xⁿ terms, compute the value as (bⁿ⁺¹ − aⁿ⁺¹) ÷ (n + 1) directly, term by term. This avoids writing the full bracket.
When the lower limit is 0, the lower substitution is usually 0 for polynomial terms. Only substitute the upper limit.
For extra cost from a to b, do not find the constant. Just compute the definite integral of MC.
For surplus, find x₀ and p₀ first. Then do the integral and subtract the rectangle p₀·x₀.
Use the options to reject answers with the wrong sign or an obvious unit mismatch, then verify the closest option by a quick recomputation.
If the integral needs a long substitution and the options are close, skip it. A wrong answer costs 0.25 marks.

Common mistakes in Definite Integrals and Their Applications

Adding the constant C in a definite integral.
Students carry the habit from indefinite integrals.
Fix: With limits, C cancels in F(b) − F(a). Leave it out.
Subtracting in the wrong order, F(a) − F(b).
Rushing, or limits written in a different order than expected.
Fix: Always do upper limit minus lower limit. If the limits are reversed, the answer is negative.
Using the power rule on 1/x and getting x⁰ ÷ 0.
Students forget the n = −1 exception.
Fix: Remember ∫ (1/x) dx = log x. Rewrite 1/x² as x⁻² and use the power rule only for n ≠ −1.
Finding total cost from MC without adding fixed cost.
Students stop after integrating and forget the constant has meaning.
Fix: Use C(x) = ∫ MC dx + fixed cost. Only the extra cost between two outputs ignores fixed cost.
Computing consumer surplus as just the integral of demand.
Students forget to subtract the amount actually paid.
Fix: CS = ∫ D(x) dx from 0 to x₀ minus p₀·x₀. For PS, it is p₀·x₀ minus the integral of supply.
Finding area directly when the curve crosses the x-axis.
Positive and negative parts cancel in the integral.
Fix: Split at the crossing point and add the absolute values of the parts.

Worked examples

Example 1

The value of ∫ from 1 to 2 of (3x² + 2x) dx is: (a) 8 (b) 10 (c) 12 (d) 6

Show the solution

Integrate: ∫ (3x² + 2x) dx = x³ + x².
Upper limit x = 2: 8 + 4 = 12.
Lower limit x = 1: 1 + 1 = 2.
Subtract: 12 − 2 = 10.

Answer: (b) 10

Example 2

The marginal cost of a firm is MC = 6x + 4, and fixed cost is ₹100. The total cost of producing 10 units is: (a) ₹440 (b) ₹340 (c) ₹540 (d) ₹240

Show the solution

Total cost C(x) = ∫ (6x + 4) dx + k = 3x² + 4x + k.
Fixed cost means C(0) = 100, so k = 100.
C(x) = 3x² + 4x + 100.
C(10) = 3(100) + 40 + 100 = 300 + 40 + 100 = 440.

Answer: (a) ₹440

Example 3

The demand curve is p = 20 − x and the supply curve is p = 2 + x. The consumer surplus at equilibrium is: (a) ₹81 (b) ₹40.5 (c) ₹162 (d) ₹18

Show the solution

Equilibrium: 20 − x = 2 + x, so 2x = 18 and x₀ = 9.
Equilibrium price p₀ = 20 − 9 = 11.
CS = ∫ from 0 to 9 of (20 − x) dx − p₀·x₀.
∫ (20 − x) dx = 20x − x²/2. At 9: 180 − 40.5 = 139.5. At 0: 0.
Amount paid = 11 × 9 = 99.
CS = 139.5 − 99 = 40.5.

Answer: (b) ₹40.5

Exam tips

Scan for zero-answer shortcuts: equal limits, or an odd function over symmetric limits.
In cost questions, check whether the question asks for total cost or extra cost between two outputs. Only total cost needs fixed cost.
For surplus, always find the equilibrium point first. Many wrong options are built from forgetting to subtract p₀·x₀.
With linear demand and supply, surplus equals the area of a triangle. Use ½ × base × height as a cross-check.
Do not spend more than about a minute on one integral. Skip if the algebra is long.

Practice questions from Differential and Integral Calculus

Definite Integrals and Their Applications: frequently asked questions

What is the difference between definite and indefinite integrals?

An indefinite integral gives a function plus a constant C. A definite integral has limits and gives one number, found as F(b) − F(a). The constant cancels in the subtraction.

How do I find total cost from marginal cost using integration?

Integrate MC with respect to output to get C(x) plus a constant. Use the fixed cost to find the constant, since C(0) equals fixed cost. Then substitute the required output.

Can a definite integral be negative?

Yes. If the curve lies below the x-axis between the limits, the integral is negative. Area is always positive, so take the absolute value when the question asks for area.

How are consumer and producer surplus calculated?

Find the equilibrium by equating demand and supply prices. Consumer surplus is the integral of demand from 0 to x₀ minus p₀x₀. Producer surplus is p₀x₀ minus the integral of supply from 0 to x₀.