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MHT CET202523 Apr 2025Evening ShiftMathematicsArea Under CurvesActual

The area bounded by the curve y =x^2+3, y =x, x=3 and y -axis is...

Options

  1. A9 2 sq. units
  2. B18 sq. units
  3. C27 2 sq. units
  4. D27 3 sq. units

Correct answer

C. 27 2 sq. units

Step-by-step solution

Area Between Curves The area bounded by y=x^2+3 , y=x , x=3 , and x=0 is determined by the definite integral of the difference between the upper and lower functions over the interval [0, 3] . Since x^2+3 > x for all x (as x^2 - x + 3 has discriminant -11 and is always positive), the area is computed as: A = ₀^3 [(x^2+3) - x] dx = ₀^3 (x^2 - x + 3) dx Evaluating the integral: A = [ x^3 3 - x^2 2 + 3x ]₀^3 = ( 27 3 - 9 2 + 9 ) - (0) = 9 - 9 2 + 9 = 27 2 The area is 27 2 square units, corresponding to option C .

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