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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|.

The given region is
   $open curly brackets left parenthesis straight x comma space straight y right parenthesis space colon space straight x squared space less or equal than straight y less or equal than open vertical bar straight x close vertical bar close curly brackets$
This region is the intersection of the following regions
   $straight R subscript 1 space equals space open curly brackets left parenthesis straight x comma space straight y right parenthesis space colon space straight x squared space less or equal than space straight y close curly brackets straight R subscript 2 space equals open curly brackets open parentheses straight x comma space straight y close parentheses colon space straight y space less or equal than open vertical bar straight x close vertical bar close curly brackets$
Consider the equations
   $straight y equals straight x squared$ ...(1)
y = x ...(2)
and y = -x ...(3)

From (1) and (2), we get
   $straight x equals straight x squared space space space space space space space space space rightwards double arrow space space space straight x squared minus straight x space equals space 0 space space space space space space space space rightwards double arrow space space space straight x left parenthesis straight x minus 1 right parenthesis space space space space space space rightwards double arrow space space space space space straight x space equals space 0 comma space space 1$
from (2), y = 0, 1
∴ curve (1) and (2) intersect in the points O (0, 0) and A (1, 1).
Similarly, curves (1) and (3) intersect in the points O (0, 0) and B (-1, 1)
Required area = area of shaded region = 2 (area of region OAO)
$equals 2 open square brackets integral subscript 0 superscript 1 straight x space dx minus integral subscript 0 superscript 1 straight x squared space dx close square brackets space equals space 2 space open curly brackets open square brackets straight x squared over 2 close square brackets subscript 0 superscript 1 minus open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 1 close curly brackets equals space 2 open curly brackets open parentheses 1 half minus 0 close parentheses minus open parentheses 1 third minus 0 close parentheses close curly brackets space equals 2 open parentheses 1 half minus 1 third close parentheses space equals space 2 cross times 1 over 6 equals 1 third space sq space. units.$

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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.

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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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