Integrals

Maths · Class 12

Simulation · Maths · Class 12

The area function

"The area function" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "The definite integral and the fundamental theorem" in Integrals. The shaded area from 0 up to x is A(x). The small graph below plots A(x) as x changes. Watch its tangent: the steepness of A at x always equals the height of the curve at x. A(x) = ∫₀ˣ f(t) dt has A′(x) = f(x): the first fundamental theorem. So any antiderivative F gives the area as F(b) − F(a). JEE Main asked 73 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson The definite integral and the fundamental theorem in Integrals. Change the values and watch what happens.

The area functionMaths · Class 12

In JEE Main

42 shifts analysed

73 questions from Integrals across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §7.7–7.8

  • A definite integral ∫ₐᵇ f(x) dx has a single numerical value. a is the lower limit and b the upper limit.
  • For f(x) ≥ 0 on [a, b], it is the area of the region between y = f(x), the x-axis and the lines x = a and x = b.
  • Area function: fix the left end at a and let the right end move. A(x) = ∫ₐˣ f(t) dt is the area up to x, so it is itself a function of x.
  • First fundamental theorem: if f is continuous on [a, b], then A′(x) = f(x). The area grows at a rate equal to the height of the curve at the moving edge.
  • Second fundamental theorem: if f is continuous on [a, b] and F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a), written [F(x)]ₐᵇ.
  • The constant C cancels in F(b) − F(a), so it is left out when evaluating a definite integral.
  • The integrand must be continuous on the whole closed interval. If it is not, the theorem cannot be applied straight across the bad point.
  • Worked example: ∫₂³ x² dx = [x³/3] from 2 to 3 = 27/3 − 8/3 = 19/3.
  • Worked example: x/((x + 1)(x + 2)) = −1/(x + 1) + 2/(x + 2), so ∫₁² x/((x + 1)(x + 2)) dx = [2 log(x + 2) − log(x + 1)] from 1 to 2 = log(32/27).
  • Worked example: ∫₀^(π/4) sin³2t cos 2t dt = [sin⁴2t / 8] from 0 to π/4 = 1/8.

Questions

3 answers

What does the "The area function" simulation show?
The shaded area from 0 up to x is A(x). The small graph below plots A(x) as x changes. Watch its tangent: the steepness of A at x always equals the height of the curve at x.
Does JEE Main ask questions from Integrals?
Yes. JEE Main asked 73 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 "The definite integral and the fundamental theorem" in Integrals (Class 12 Maths).

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