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COMEDK2025MathematicsArea Under CurvesActual

The area of the region bounded by the curve y^2=2 y-x and the y -axis is

Options

  1. A1 sq unit
  2. B20 3 sq units
  3. C2 sq units
  4. D4 3 sq units

Correct answer

D. 4 3 sq units

Step-by-step solution

The given equation of the curve is y^2 = 2y - x , which can be rewritten as x = 2y - y^2 . The curve intersects the y -axis where x = 0 . Setting x = 0 in the equation x = 2y - y^2 , we get 2y - y^2 = 0 , which implies y(2 - y) = 0 . Thus, the points of intersection are y = 0 and y = 2 . The area A of the region bounded by the curve and the y -axis is given by the integral of x with respect to y from y = 0 to y = 2 . A = ₀² (2y - y^2) dy Evaluating the integral: A = [ y^2 - y^3 3 ]₀² A = ( 2^2 - 2^3 3 ) - (0 - 0) A

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