MathsMediumClass 12

Definite Integration as Limit of Sum

Definite Integration & Area

JEE Qs

Hard

min

Master the pattern recognition for converting summation index r/n to x and 1/n to dx, and accurately determining integration limits based on the summation range.

🧮 Key Formulas

∫(a to b) f(x) dx = lim (n→∞) [(b-a)/n * ∑(r=1 to n) f(a + r*(b-a)/n)]

If a=0 and b=1, then ∫(0 to 1) f(x) dx = lim (n→∞) [1/n * ∑(r=1 to n) f(r/n)]

General transformation rule: Replace r/n with x, 1/n with dx, and lim (n→∞) ∑ with ∫. Lower limit = lim (n→∞) (lower index of r / n). Upper limit = lim (n→∞) (upper index of r / n).

✅ Key Points for JEE

1The core idea is to recognize expressions of the form lim (n→∞) [ (1/n) * ∑ f(r/n) ] or similar variations, and convert them into a definite integral.
2The limits of integration (upper and lower) are determined by taking the limit of the ratio (index of summation / n) as n approaches infinity for both the starting and ending values of the index 'r'.
3Ensure the expression is in the form of f(r/n) * (1/n) inside the summation before converting. Sometimes algebraic manipulation (e.g., factoring out n, dividing by n) is required.
4This method is fundamental to understanding the geometric interpretation of definite integration as the area under a curve, forming the basis of Riemann sums.
5Be careful when the sum does not start from r=1 or end at r=n. For example, if the sum runs from r=0 to n-1, the lower limit remains 0 and the upper limit remains 1.

⚠️ Common Mistakes

✕Incorrectly identifying the limits of integration, especially when the summation index 'r' does not run from 1 to n (e.g., 0 to n-1, or 1 to 2n).
✕Failure to transform the expression into the standard f(r/n) * (1/n) format before replacing terms with x and dx.
✕Confusing this specific type of limit of sum with other types of limits of series that might require different techniques (e.g., L'Hopital's rule, properties of convergent series).
✕Not handling constants correctly inside or outside the summation sign during transformation.

📝 Practice Questions

See all

Q1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →

2025·NumericalHard

Q11.Let the area enclosed between the curves |y| = 1 −x2 and x2 + y2 = 1 be α. If 9α = βπ + γ; β, γ are integers, then the value of |β −γ| equals. (1) 27 (2) 33 (3) 15 (4) 18

2025·MCQMedium

Q21.If 24 ∫ 0 4 (sin 4x − 12π + [2 sin x])dx = 2π + α, where [⋅] denotes the greatest integer function, then α is equal to _______.

2025·NumericalMedium

Q6. Let for f(x) = 7 tan8 x + 7 tan6 x −3 tan4 x −3 tan2 x, I1 = ∫π/40 f(x)dx and I2 = ∫π/40 xf(x)dx. Then 7I1 + 12I2 is equal to : (1) 2 (2) 1 (3) 2π (4) π

2025·MCQMedium

Q13.The area of the region, inside the circle (x −2√3)2 + y2 = 12 and outside the parabola y2 = 2√3x is : (1) 3π + 8 (2) 6π −16 (3) 3π −8 (4) 6π −8

2025·MCQHard

Q7. The area of the region enclosed by the curves y = x2 −4x + 4 and y2 = 16 −8x is : (1) 8 (2) 4 3 3 (3) 8 (4) 5 x ∈R. Then the numbers of local maximum and local minimum points of f ,

2025·MCQMedium

NCERT Chapters

Class 12 Maths Ch 7: Integrals