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Application of Integrals

Name: Zigya - For The Curious Learner
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Long Answer Type

21.

Draw a graph of $straight x squared over 9 plus straight y squared over 25 space equals space 1$ and evaluate area bounded by it.

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22. Using definite integrals, find the area of the ellipse

straight x squared over 4 plus straight y squared over 9 space equals space 1

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23.

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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Short Answer Type

24.

Using definite integrals, find the area of the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1$ .

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25.
Sketch the region of the ellipse and find its area, using integration.
$straight x squared over straight b squared plus straight y squared over straight a squared equals 1.$

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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Long Answer Type

26. Find the area of the region in the first quadrant enclosed by the x-axis, the line

straight x equals square root of 3 space straight y

and the circle x² + y² = 4.

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27. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x, and the circle x² + y² = 32.

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28.

Find the area bounded by the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space the space$ ordinates x = a e and x = 0 where b² = a² (1 - e²) and e < 1.