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KVPY2010MathematicsArea Under Curves

The following figure shows the graph of a continuous function y = f ( x ) on the interval [1,3] . The points A , B , C have coordinates (1,1),(3,2),(2,3) respectively, and the lines L ₁ and L ₂ are parallel, with L ₁ being tangent to the curve at C . If the area under the graph of y = f ( x ) from x =1 to x =3 is 4 square units, then the area of the shaded region is-

Options

  1. A2
  2. B3
  3. C4
  4. D5

Correct answer

A. 2

Step-by-step solution

Equation of ₂ y -1= 2-1 3-1 ( x -1)2 y-2=x-12 y-x=1 Equation of ₁=y-3= 1 2 (x-2) array l 2 y-6=x-2 2 y-x=4 D (1, 5 2 ), E (3, 7 2 ) array area under f(x)=4 aligned shaded area &= area of trapezium DEFG - area under f ( x ) &= 1 2 ( 5 2 + 7 2 ) 2-4 &=6-4=2 aligned

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