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📐 Mathematics  ·  Application of Integrals  ·  JEE

The area under the curve y = f(x), above the x-axis, between x = a and x = b is given by:

Answer: Integral from a to b of f(x) dx.

  • A f(b) - f(a)
  • B Integral from a to b of f(x) dx
  • C f(a) + f(b)
  • D Derivative of f at b minus derivative at a

Correct answer: B. Integral from a to b of f(x) dx

Explanation: The area bounded by a non-negative curve, the x-axis, and the vertical lines x=a, x=b is the definite integral from a to b of f(x) dx.

Area Under a Curve = Definite Integralxyx=ax=bArea = ∫ₐᵇ f(x)dx

The definite integral ∫ₐᵇf(x)dx computes the exact area of the shaded region bounded by the curve, the x-axis, and the vertical lines x=a and x=b - the same idea behind the Riemann sum, but evaluated exactly rather than approximated by rectangles.

Concept context

Using definite integrals to compute the area under a curve, the area between two curves, and the areas enclosed by standard curves like circles, parabolas, and ellipses.

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