Question

Sketch the region of the ellipse and find its area, using integration.
$straight x squared over straight b squared plus straight y squared over straight a squared equals 1.$

Solution

The equation of ellipse is
$straight x squared over straight b squared plus straight y squared over straight a squared space equals space 1 space space space or space space straight y squared over straight a squared space equals space 1 minus straight x squared over straight b squared space space or space straight y squared space equals space straight a squared over straight b squared left parenthesis straight b squared minus straight x squared right parenthesis$
$therefore space space space space straight y space equals space straight a over straight b square root of straight b squared minus straight x squared end root space space space left parenthesis in space the space first space quadrant right parenthesis$ ...(1)
The ellipse is symmetrical about both the axes.
∴ required area = 4 (area OAB)
$equals space 4 integral subscript straight a superscript straight b straight y space dx space equals space 4 space integral subscript straight a superscript straight b straight a over straight b square root of straight b squared minus straight x squared end root space dx$
$equals space fraction numerator 4 straight a over denominator straight b end fraction integral subscript 0 superscript straight b square root of straight b squared minus straight x squared end root space dx$ $open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$
$equals space fraction numerator 4 straight a over denominator straight b end fraction open square brackets fraction numerator straight x square root of straight b squared minus straight x squared end root over denominator 2 end fraction plus straight b squared over 2 sin to the power of negative 1 end exponent open parentheses straight x over straight b close parentheses close square brackets subscript 0 superscript straight b equals space fraction numerator 4 straight a over denominator straight b end fraction open square brackets open curly brackets fraction numerator straight b square root of straight b squared minus straight b squared end root over denominator 2 end fraction plus straight b squared over 2 sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis close curly brackets minus open curly brackets 0 plus straight b squared over 2 sin to the power of negative 1 end exponent 0 close curly brackets close square brackets equals space fraction numerator 4 straight a over denominator straight b end fraction open square brackets open curly brackets 0 plus straight b squared over 2. straight pi over 2 close curly brackets minus open curly brackets 0 plus 0 close curly brackets close square brackets space equals space fraction numerator 4 straight a over denominator straight b end fraction. fraction numerator straight pi space straight b squared over denominator 4 end fraction space equals space straight pi space straight a space straight b space sq. space units. space$

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Vector Algebra

Sketch the region of the ellipse and find its area, using integration.
$straight x squared over straight b squared plus straight y squared over straight a squared equals 1.$

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