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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}

Given region is
{(x, y): y² ≤ 8 x, x² +y² ≤ 9}
Consider the equations
y² = 8 x    ...(1)
and x² + y² = 9    ...(2)
From (1) and (2). we get,
x² + 8 x = 9 or x² + 8 x-9 = 0⇒ (x + 9)(x - 1) = 0
⇒    x = - 9, 1
which gives the abscissa of the points of intersection P and Q.
Rejecting negative value of x, we get, x = 1
Required area = Area of shaded region
= 2 (area of region APOA)
= 2[area of region AMPA + area of region MOPM]
$equals space 2 open square brackets integral subscript 0 superscript 1 2 square root of 2 square root of straight x dx plus integral subscript 1 superscript 3 square root of 9 minus straight x squared end root dx close square brackets equals space 4 square root of 2 integral subscript 0 superscript 1 straight x to the power of 1 half end exponent dx plus 2 integral subscript 1 superscript 3 square root of left parenthesis 3 right parenthesis squared minus straight x squared end root space dx space equals space 4 square root of 2 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 plus 2 open square brackets fraction numerator straight x square root of left parenthesis 3 right parenthesis squared 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 1 superscript 3 equals space fraction numerator 8 square root of 2 over denominator 3 end fraction open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 1 plus space open square brackets straight x square root of 9 minus straight x squared end root plus 9 space s i n to the power of negative 1 end exponent open parentheses straight x over 3 close parentheses close square brackets subscript 1 superscript 3 equals space fraction numerator 8 square root of 2 over denominator 3 end fraction left parenthesis 1 minus 0 right parenthesis plus open parentheses 3 square root of 9 minus 9 end root plus 9 space sin to the power of negative 1 end exponent 1 close parentheses minus open parentheses 1 square root of 9 minus 1 end root plus 9 space sin to the power of negative 1 end exponent 1 third close parentheses equals space fraction numerator 8 square root of 2 over denominator 3 end fraction plus open parentheses 0 plus 9 cross times straight pi over 2 close parentheses minus open parentheses square root of 8 plus 9 space sin to the power of negative 1 end exponent 1 third close parentheses equals space fraction numerator 8 square root of 2 over denominator 3 end fraction plus fraction numerator 9 straight pi over denominator 2 end fraction minus 2 square root of 2 minus 9 space sin to the power of negative 1 end exponent open parentheses 1 third close parentheses equals space open square brackets fraction numerator 2 square root of 2 over denominator 3 end fraction plus fraction numerator 9 straight pi over denominator 2 end fraction minus 9 space sin to the power of negative 1 end exponent open parentheses 1 third close parentheses close square brackets space sq. space units.$

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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.

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70. Draw a rough sketch of the region {(x, y): y² ≤ 5 x, 5 x² + 5 y² ≤ 36} and find the area enclosed by the region using method of integration.

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