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Application of Integrals

Name: Zigya - For The Curious Learner
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Long Answer Type

71. Draw a rough sketch of the region {(x, y): y² ≤ 3 x, 3 x² + 3 y² ≤ 16} and find the area enclosed by the region using method of integration.

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72.

Find the area of the smaller part of the circle x² + y² = a cut oil by the line $straight x equals fraction numerator straight a over denominator square root of 2 end fraction$

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73. Find the area of the smaller region bounded by the ellipse

straight x squared over 9 plus straight y squared over 4 space equals space 1

and the straight line

straight x over 3 plus straight y over 2 space equals space 1

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74.

Using integration, find the area of the smaller region bounded by the curve $straight x squared over 16 plus straight y squared over 9 space equals space 1$ and the straight line $straight x over 4 plus straight y over 3 equals 1.$

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75.
Find the area of smaller region bounded by the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared space equals 1$ and the straight line $straight x over straight a plus straight y over straight b equals 1$

Consider the equations
$straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1$ ...(1)
and $straight x over straight a plus straight y over straight b equals 1$ ...(2)

Line (2) meets ellipse (1) in A (a, 0) and B (0, b)
Area of shaded region = Area of region OAB area of ∆OAB
$equals space straight b over straight a integral subscript 0 superscript straight a square root of straight a squared minus straight x squared end root space dx minus straight b over straight a integral subscript 0 superscript straight a left parenthesis straight a minus straight x right parenthesis space dx equals space straight b over straight a open square brackets fraction numerator straight x square root of straight a squared minus straight x squared end root over denominator 2 end fraction plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets subscript 0 superscript straight a minus space fraction numerator straight b over denominator 2 straight a end fraction open square brackets fraction numerator left parenthesis straight a minus straight x right parenthesis squared over denominator 1 end fraction close square brackets subscript 0 superscript straight a equals space straight b over straight a open square brackets open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 1 close parentheses minus open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets plus fraction numerator straight b over denominator 2 straight a end fraction left square bracket 0 minus straight a squared right square bracket equals space straight b over straight a open square brackets straight a squared over 2 cross times straight pi over 2 minus 0 minus 0 close square brackets space minus space ab over 2 space equals space πab over 4 minus ab over 2 space equals space ab over 4 left parenthesis straight pi minus 2 right parenthesis space sq. space units.$
which is required area.

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76. Evaluate

integral subscript 0 superscript straight r square root of straight r squared minus straight x squared end root dx

, where x is a fixed positive number, Hence, prove that the area of a circle of radius r is

straight pi space straight r squared.

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Short Answer Type

77.

Find the whole area of the circle x² + y² = a².

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Multiple Choice Questions

78. Area lying in the first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2 is

$straight pi$
$straight pi over 2$
$straight pi over 3$
$straight pi over 3$

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79. Area of the region bounded by the curve y² = 4 x, y-axis and the line y = 3 is

2
$9 over 4$
$9 over 3$
$9 over 3$

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80. Smaller area enclosed by the circle x² + y² = 4 and the line x + y = 2 is
Choose the correct answer.
or
Draw the rough sketch and find the area of the region:
{(x, y): x² + y² < 4, x + y > 2}.