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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.

The equation of ellipse is
$straight x squared over 9 plus straight y squared over 25 space equals space 1$
or $straight y squared over 25 space equals space 1 minus straight x squared over 9$
or $straight y squared space equals space 25 over 9 left parenthesis 9 minus straight x squared right parenthesis$

$therefore space space space space space straight y space equals space 5 over 3 square root of 9 minus straight x squared end root$ ...(1)
(in the first quadrant)
The ellipse is symmetrical about both the axes.
∴ required area = 4 (area AOB)
$equals space 4 integral subscript 0 superscript 3 straight y space dx space equals space 4 integral subscript 0 superscript 3 5 over 3 square root of 9 minus straight x squared end root space dx$ $open square brackets because space of space left parenthesis 1 right parenthesis close square brackets space space$
$equals space 20 over 3 integral square root of left parenthesis 3 right parenthesis squared minus straight x squared end root space dx space equals space 20 over 3 open square brackets fraction numerator straight x square root of 9 minus straight x squared end root over denominator 2 end fraction plus fraction numerator left parenthesis 3 right parenthesis squared over denominator 2 end fraction sin to the power of negative 1 end exponent open parentheses straight x over 3 close parentheses close square brackets subscript 0 superscript 3 equals space 20 over 3 open square brackets open parentheses fraction numerator 3 square root of 9 minus 9 end root over denominator 2 end fraction plus 9 over 2 sin to the power of negative 1 end exponent 1 close parentheses minus open parentheses 0 plus 9 over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets equals space 20 over 3 open square brackets open parentheses 0 plus 9 over 2 cross times straight pi over 2 close parentheses minus left parenthesis 0 plus 0 right parenthesis close square brackets space equals space 20 over 3 cross times fraction numerator 9 straight pi over denominator 4 end fraction space equals space 15 space straight pi space sq. space units.$

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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

148 Views

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.

112 Views

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.