Question

Draw a graph of $straight x squared over 9 plus fraction numerator straight y squared over denominator 16 space end fraction space equals space 1$ and evaluate area bounded by it.

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Solution

The equation of ellipse is

straight x squared over 9 plus straight y squared over 16 space equals space 1

straight y squared over 16 space equals space 1 minus straight x squared over 9

straight y squared space equals space 16 over 9 left parenthesis 9 minus straight x squared right parenthesis

therefore space space space space space space space straight y space equals space 4 over 3 square root of 9 minus straight x squared end root

...(1)
(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 integral subscript 0 superscript 3 4 over 3 square root of 9 minus straight x squared end root space dx

[

because space of space left parenthesis 1 right parenthesis right square bracket

equals space 16 over 3 integral subscript 0 superscript 3 square root of left parenthesis 3 right parenthesis squared minus straight x squared end root space dx space equals space 16 over 3 open square brackets fraction numerator straight x square root of 9 minus straight x squared end root over denominator 2 end fraction plus fraction numerator left parenthesis 3 right parenthesis squared over denominator 2 end fraction 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 16 over 3 open square brackets open parentheses 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 1 close parentheses minus space open parentheses 0 plus 9 over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets equals space 16 over 3 open square brackets open parentheses 0 plus 9 over 2 cross times straight pi over 2 close parentheses minus left parenthesis 0 plus 0 right parenthesis close square brackets space equals space 16 over 3 cross times fraction numerator 9 straight pi over denominator 4 end fraction equals space space 12 space straight pi space sq. space units.

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