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Easy

Question

The area of the plane region bounded by the curves x plus 2 y to the power of 2 end exponent equals 0 and x plus 3 y to the power of 2 end exponent equals 1 is equal to

  1. fraction numerator 5 over denominator 3 end fraction    
  2. fraction numerator 1 over denominator 3 end fraction    
  3. fraction numerator 2 over denominator 3 end fraction    
  4. fraction numerator 4 over denominator 3 end fraction    

The correct answer is: fraction numerator 4 over denominator 3 end fraction


    Solving the equations, we get the points of intersection as
    left parenthesis negative 21 right parenthesis and left parenthesis negative 2 comma negative 1 right parenthesis. T h e bounded region is shown a shaded region in Fig 24.29.

    Figure 24.29
    The required area is
    2 not stretchy integral subscript 0 end subscript superscript 1 end superscript left square bracket left parenthesis right square bracket minus 3 y to the power of 2 end exponent right parenthesis minus left parenthesis negative 2 y to the power of 2 end exponent right parenthesis right square bracket d y equals 2 not stretchy integral subscript 0 end subscript superscript 1 end superscript left parenthesis 1 minus y to the power of 2 end exponent right parenthesis d y equals 2 left square bracket y minus fraction numerator y to the power of 3 end exponent over denominator 3 end fraction right square bracket subscript 0 end subscript superscript 1 end superscript
    equals 2 cross times fraction numerator 2 over denominator 3 end fraction equals fraction numerator 4 over denominator 3 end fraction s q units
    Hence, the correct answer is option (D).

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