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KCET2026MathematicsArea Under Curves

The area of the region bounded by the curve y^2 = x^3 , the y -axis and the lines y = 1 and y = 8 is

Options

  1. A155 3 sq. units
  2. B93 5 sq. units
  3. C93 sq. units
  4. D155 sq. units

Correct answer

B. 93 5 sq. units

Step-by-step solution

The equation of the curve is given by y^2 = x^3 , which can be written as x = y^ 2/3 . The required area is bounded by the curve x = y^ 2/3 , the y -axis ( x = 0 ), and the horizontal lines y = 1 and y = 8 . The area A is given by the definite integral with respect to y : A = ₁⁸ x , dy Substituting x = y^ 2/3 : A = ₁⁸ y^ 2/3 , dy Integrating the function: A = [ y^ 5/3 5/3 ]₁⁸ A = 3 5 [ y^ 5/3 ]₁⁸ Evaluating the limits: A = 3 5 ( 8^ 5/3 - 1^ 5/3 ) Since 8^ 5/3 = (2^3)^ 5/3 = 2^5 = 32 and 1^ 5/3 = 1 : A = 3 5 (32 - 1

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