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📐 Mathematics  ·  Integrals  ·  JEE

Integral from a to b of f(x) dx = - integral from b to a of f(x) dx. This is because:

Answer: Reversing limits changes sign.

  • A Reversing limits changes sign
  • B f(x) is odd
  • C Integration is linear
  • D The antiderivative changes

Correct answer: A. Reversing limits changes sign

Explanation: By the Fundamental Theorem: ∫ₐᵇf(x)dx = F(b)−F(a). Reversing limits: ∫ᵦₐf(x)dx = F(a)−F(b) = −[F(b)−F(a)]. Reversing integration limits negates the value; this property is independent of whether f is odd or even.

xyabArea = integral of f(x) dxRectangles approximate the area (Riemann sum)

The exact area under the curve from a to b (shaded) is approximated by a few rectangles, the Riemann sum idea behind the definite integral.

Concept context

Indefinite and definite integrals, areas under curves

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