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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.

The equation of circle is
$straight x squared plus straight y squared space equals space 4$ ...(1)
The equation of line is
   $straight x space equals square root of 3 space straight y$ ...(2)
From (1) and (2), we get,
   $3 straight y squared plus straight y squared space equals space 4 space space space space or space space space space 4 straight y squared space equals space 4$
$therefore space space space space space space space space straight y squared space equals space 1 rightwards double arrow space space space space space space straight y space equals space 1 space space space space space space space space space rightwards double arrow space space space space space straight x space equals space square root of 3$

∴ point of intersection of circle (1) and line (2) is P $left parenthesis square root of 3 comma space 1 right parenthesis$ .
From P, draw PM ⊥ x-axis.
Also OA = radius of circle = 2
∴ A is (2, 0)
Required area = Area OMAPO = Area of ∆OMP + area MAP
= A₁ + A₂    ..(1 )
where $straight A subscript 1 space equals space area space of space increment OMP space equals space fraction numerator 1 over denominator square root of 3 end fraction integral subscript 0 superscript square root of 3 end superscript space straight x space dx$

equals space fraction numerator 1 over denominator square root of 3 end fraction open square brackets straight x squared over 2 close square brackets subscript 0 superscript square root of 3 end superscript space equals space left parenthesis 1 right parenthesis space fraction numerator 1 over denominator 2 square root of 3 end fraction left parenthesis 3 minus 0 right parenthesis space equals space fraction numerator square root of 3 over denominator 2 end fraction

and

straight A subscript 2 space equals space integral subscript square root of 3 end subscript superscript 2 space straight y space dx space equals space integral subscript square root of 3 end subscript superscript 2 square root of 4 minus straight x squared end root space dx space equals space integral subscript square root of 3 end subscript superscript 2 square root of left parenthesis 2 right parenthesis squared minus straight x squared end root space dx

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

therefore space space from space left parenthesis 1 right parenthesis comma space required space area space equals space fraction numerator square root of 3 over denominator 2 end fraction plus straight pi over 3 minus fraction numerator square root of 3 over denominator 2 end fraction equals space straight pi over 3 space sq. space units

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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.