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Application of Integrals

Name: Zigya - For The Curious Learner
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Short Answer Type

61.

Find the area of the region {(x, y): x² + y² ≤ 1 ≤ x + y}.

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Long Answer Type

62.

Find the area of the region bounded by the circle x² + y² = 1 and x + y = 1. Also draw a rough sketch.

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63.

Find the area of the region {(x, y): x² ≤ y ≤ |x|}.
Or
Find the area of the region bounded by the parabola y = x² and y = |x|.

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64. Draw the rough sketch and find the area of the region:
{(x, y) : y² ≤ 8 x, x² + x² ≤ 9}

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65. Find the area of the region {(x, y): y² ≤ 4 x, 4x² + 4 y² ≤ 9}

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66. Find the area lying above x-axis and included between the circle x² + y² = 8 x and inside of the parabola y² = 4 x.

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67. Calculate the area enclosed in the region:

open curly brackets left parenthesis straight x comma space straight y right parenthesis space semicolon space space straight x squared plus straight y squared space less or equal than space 1 space less than space straight x plus 1 half straight y close curly brackets

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68. Find the area of the circle x² + y² = 16 which is exterior to the parabola y² = 6x.

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69. Find the area of the circle 4x² + 4y² = 9 which is interior to the parabola y²= 4 x.

Consider the equations
$straight y squared space equals space 4 straight x$ ...(1)
and $4 straight x squared plus 4 straight y squared equals 9$
i.e., $straight x squared plus straight y squared space equals space 9 over 4$ ...(2)
From (1) and (2), we get,
$straight x squared plus 4 straight x equals 9 over 4 space or space space 4 straight x squared plus 16 straight x minus 9 space equals space 0$

$therefore space space space straight x space equals space fraction numerator negative 16 plus-or-minus square root of 256 plus 144 end root over denominator 8 end fraction equals fraction numerator negative 16 plus-or-minus 20 over denominator 8 end fraction equals space 1 half comma space minus 9 over 2$

$therefore space space space space from space left parenthesis 1 right parenthesis comma space space space straight y squared space equals space 1 half$ $open square brackets because space space straight y squared equals negative 9 over 2 space does space not space give space real space points close square brackets$

$therefore space space space space straight y space equals space fraction numerator 1 over denominator square root of 2 end fraction comma space minus fraction numerator 1 over denominator square root of 2 end fraction$

$therefore space space space space curve space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space in space the space points$
$straight P open parentheses 1 half comma space fraction numerator 1 over denominator square root of 2 end fraction close parentheses space and space straight Q space equals space open parentheses 1 half comma space minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses$

From P, draw $PM space perpendicular space straight x minus axis$
Here $OA space equals space 3 over 2$
Required area = Area of shaded region
= 2 (area of region OAPO) = 2 [area of region OMPO + area of region MAPM]

$equals space 2 open square brackets integral subscript 0 superscript 1 divided by 2 end superscript 2 square root of straight x dx plus integral subscript 1 divided by 2 end subscript superscript 3 divided by 2 end superscript square root of 9 over 4 minus straight x squared end root close square brackets space equals space 4 integral subscript 0 superscript 1 divided by 2 end superscript straight x to the power of 1 divided by 2 end exponent dx plus 2 integral subscript 1 divided by 2 end subscript superscript 3 divided by 2 end superscript square root of open parentheses 3 over 2 close parentheses squared minus straight x squared end root dx equals space 4 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 1 half end superscript plus 2 open square brackets fraction numerator straight x square root of begin display style 9 over 4 minus straight x squared end style end root over denominator 2 end fraction plus open parentheses begin display style 3 over 2 end style close parentheses squared over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator 3 divided by 2 end fraction close parentheses close square brackets subscript 1 half end subscript superscript 3 over 2 end superscript equals space 8 open square brackets straight x to the power of 3 divided by 2 end exponent close square brackets subscript 0 superscript 1 divided by 2 end superscript space plus space 2 space open square brackets 1 half straight x square root of 9 over 4 minus straight x squared end root plus 9 over 8 sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 3 end fraction close parentheses close square brackets subscript 1 half end subscript superscript 3 over 2 end superscript equals space 8 over 3 open square brackets open parentheses 1 half close parentheses to the power of 3 over 2 end exponent minus 0 close square brackets plus space 2 open square brackets open parentheses 0 plus 9 over 8 sin to the power of negative 1 end exponent 1 close parentheses minus open curly brackets 1 fourth square root of 9 over 4 minus 1 fourth end root plus 9 over 8 sin to the power of negative 1 end exponent 1 third close curly brackets close square brackets equals space 8 over 3 fraction numerator 1 over denominator 2 square root of 2 end fraction plus 2 open square brackets 9 over 8 cross times straight pi over 2 minus fraction numerator 1 over denominator 2 square root of 2 end fraction minus 9 over 8 sin to the power of negative 1 end exponent open parentheses 1 third close parentheses close square brackets equals space fraction numerator 4 over denominator 3 square root of 2 end fraction plus fraction numerator 9 straight pi over denominator 8 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction minus 9 over 4 sin to the power of negative 1 end exponent open parentheses 1 third close parentheses equals space open parentheses fraction numerator 4 square root of 2 over denominator 6 end fraction minus fraction numerator square root of 2 over denominator 2 end fraction close parentheses plus fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 sin to the power of negative 1 end exponent open parentheses 1 third close parentheses space equals space fraction numerator square root of 2 over denominator 6 end fraction plus fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 sin to the power of negative 1 end exponent open parentheses 1 third close parentheses$