The area bounded between the parabolas x²=y/4 and x² = 9y, and the straight line y = 2 is

$20 square root of 2$
$fraction numerator 10 space square root of 2 over denominator 3 end fraction$
$fraction numerator 20 space square root of 2 over denominator 3 end fraction$
$10 square root of 2$

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Solution

fraction numerator 20 space square root of 2 over denominator 3 end fraction

Required space area space equals space 2 integral subscript 0 superscript 2 space open parentheses 3 square root of straight y space minus fraction numerator square root of straight y over denominator 2 end fraction close parentheses space dy space equals space 2 integral subscript 10 superscript 2 open parentheses 5 over 2 square root of straight y close parentheses space dy equals space open square brackets fraction numerator straight y to the power of 3 divided by 2 end exponent over denominator 3 divided by 2 end fraction close square brackets subscript straight y equals 0 end subscript superscript straight y space equals 2 end superscript space equals space 10 over 3 space left parenthesis 2 to the power of 3 divided by 2 end exponent minus 0 right parenthesis space equals space fraction numerator 20 square root of 2 over denominator 3 end fraction

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