Adding up thin strips
"Adding up thin strips" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Area from thin strips" in Application of Integrals. Each blue strip is as tall as the curve at its middle, so its area is height × width. Their total is an estimate of the area under the curve. Watch the gap to the exact area as the strips get thinner. Area = the limit of Σ y Δx as the strips get thin = ∫ₐᵇ y dx. The strip dA = y dx stands for every strip at once. JEE Main asked 39 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Area from thin strips in Application of Integrals. Change the values and watch what happens.
In JEE Main
42 shifts analysed
39 questions from Application of Integrals across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §8.1–8.2
- Formulas from geometry measure figures with straight edges and circles. A region with a curved boundary needs integral calculus.
- Take the region bounded by y = f(x) (with f ≥ 0), the x-axis and the vertical lines x = a and x = b. Slice it into thin vertical strips. A strip at position x has height y = f(x) and width dx, so its area is dA = y dx.
- Adding all the strips from x = a to x = b and letting their width shrink gives the area: A = ∫ₐᵇ y dx = ∫ₐᵇ f(x) dx.
- The strip dA = y dx is called the elementary area. It sits at an arbitrary x between a and b and stands for every strip at once.
- Worked example: the region under y = x² from x = 0 to x = 3 has area ∫₀³ x² dx = [x³/3] from 0 to 3 = 9.
- Worked example: the region under y = √x from x = 0 to x = 4 has area ∫₀⁴ x^(1/2) dx = (2/3) × 4^(3/2) = 16/3.
- Worked example: under y = x² + 1 between x = 1 and x = 2 the area is [x³/3 + x] from 1 to 2 = 14/3 − 4/3 = 10/3.
- With a finite number of strips the sum only approximates the area; more and thinner strips bring it closer. The integral is the exact value those sums approach.
Questions
3 answers
- What does the "Adding up thin strips" simulation show?
- Each blue strip is as tall as the curve at its middle, so its area is height × width. Their total is an estimate of the area under the curve. Watch the gap to the exact area as the strips get thinner.
- Does JEE Main ask questions from Application of Integrals?
- Yes. JEE Main asked 39 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Area from thin strips" in Application of Integrals (Class 12 Maths).
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