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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.
$straight x squared over straight b squared plus straight y squared over straight a squared equals 1.$

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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.

The equation of ellipse
$straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1$

or $straight y squared over straight b squared space equals space 1 minus straight x squared over straight a squared$

or $straight y squared space equals space straight b squared over straight a squared left parenthesis straight a squared minus straight x squared right parenthesis$

or $straight y space equals space straight b over straight a square root of straight a squared minus straight x squared end root$ (in the first quadrant)

Ordinates are x = 0, x = a e.
Required area = $2 space integral subscript 0 superscript straight a space straight e end superscript space straight y space dx$ [ $because$ ellipse is symmetrical about x-axis]
$equals space fraction numerator 2 straight b over denominator straight a end fraction integral subscript 0 superscript ae square root of straight a squared minus straight x squared end root dx space equals space fraction numerator 2 straight b over denominator straight a end fraction open square brackets fraction numerator straight x square root of straight a squared minus straight x squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets subscript 0 superscript straight a space straight e end superscript equals space fraction numerator 2 straight b over denominator straight a end fraction open square brackets open curly brackets fraction numerator ae square root of straight a squared minus straight a squared straight e squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent open parentheses ae over straight a close parentheses close curly brackets minus open curly brackets 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close curly brackets close square brackets equals space straight b over straight a open square brackets ae. straight a square root of 1 minus straight e squared end root plus straight a squared sin to the power of negative 1 end exponent left parenthesis straight e right parenthesis close square brackets space equals space ab space open parentheses straight e square root of 1 minus straight e squared end root plus sin to the power of negative 1 end exponent straight e close parentheses$