Find the area of the region bounded by the ellipse $straight x squared over 9 plus straight y squared over 4 space equals space 1.$

The equation of ellipse is
$straight x squared over 9 plus straight y squared over 4 space equals space 1$

or $straight y squared over 4 space equals space 1 minus straight x squared over 9$

or $straight y squared space equals space 4 over 9 left parenthesis 9 minus straight x squared right parenthesis$

$therefore space space space space space straight y space equals space 2 over 3 square root of 9 minus straight x squared end root$ [In the first quadrant]

The ellipse is symmetrical about both the axes
∴ required area = 4 (area AOB)
$equals space 4 integral subscript 0 superscript 3 space straight y space dx space equals space 4 cross times 2 over 3 integral subscript 0 superscript 3 square root of 9 minus straight x squared end root space dx space equals space 8 over 3 integral subscript 0 superscript 3 square root of left parenthesis 3 right parenthesis squared minus straight x squared end root dx equals space 8 over 3 open square brackets fraction numerator straight x square root of left parenthesis 3 right parenthesis squared minus straight x squared end root over denominator 2 end fraction plus open parentheses 3 close parentheses squared over 2 sin to the power of negative 1 end exponent open parentheses straight x over 3 close parentheses close square brackets subscript 0 superscript 3 equals space 8 over 3 open square brackets open curly brackets fraction numerator 3 square root of 9 minus 9 end root over denominator 2 end fraction plus 9 over 2 sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis close curly brackets space minus space open curly brackets 0 plus 9 over 2 sin to the power of negative 1 end exponent left parenthesis 0 right parenthesis close curly brackets close square brackets equals space 8 over 3 open square brackets open curly brackets 0 plus 9 over 2 cross times straight pi over 2 close curly brackets minus open curly brackets 0 plus 9 over 2 cross times 0 close curly brackets close square brackets space equals space 8 over 3 cross times 9 over 2 cross times straight pi over 2 space equals space 6 space straight pi space sq. space units$

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