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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 $straight x over straight a plus straight y over straight b equals 1$

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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.

The equation of circle is x² + y² = a² ...(1)
Its centre is origin and radius a. We know that circle x² + y² = a² is symmetrical about both axes.

therefore space space space required space area space equals space 4 space integral subscript 0 superscript straight a straight y space dx space equals space 4 space integral subscript 0 superscript straight a square root of straight a squared minus straight x squared end root space dx

equals space 4 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 equals space 4 open square brackets open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 1 close parentheses space minus space open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets equals space 4 open square brackets open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 1 close parentheses space minus space open parentheses 0 plus straight a squared over 2 sin to the power of negative 1 end exponent 0 close parentheses close square brackets equals 4 space open square brackets straight a squared over 2 cross times straight pi over 2 close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space sin to the power of negative 1 end exponent 0 space equals space 0 close square brackets space equals space straight pi space straight a space.

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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}.