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Application of Integrals

Long Answer Type

11. Find the area of the region bounded by the curve

straight y space equals space 2 square root of 1 minus straight x squared end root

and x-axis. Draw a rough sketch..

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Short Answer Type

12. Calculate the area bounded by the parabola y² = 4 a x and its latus rectum.

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13.

Give the rough sketch of the curve y² = x and the line x = 4 and find the area between the curve and the line.

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14.

Find the area bounded between the curve y² = x and the line x = 3. Draw the rough sketch also.

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15. The area between x = y² and x = 4 is divided into two equal parts by the line x = a, find the value of a.

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16.

Find the area of the region bounded by the curve y = x² and the line y = 4.

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17.

Find the area of the region bounded by the curve y² = 4x and the line x = 3.

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18.

Using definite integrals, find the area of the circle x² + y² = a².

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Long Answer Type

19. Find the area of the region bounded by the ellipse $straight x squared over 9 plus straight y squared over 4 space equals space 1.$

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The equation of ellipse is
$straight x squared over 9 plus straight y squared over 4 space equals space 1$

or $straight y squared over 4 space equals space 1 minus straight x squared over 9$

or $straight y squared space equals space 4 over 9 left parenthesis 9 minus straight x squared right parenthesis$

$therefore space space space space space straight y space equals space 2 over 3 square root of 9 minus straight x squared end root$ [In the first quadrant]

The ellipse is symmetrical about both the axes
∴ required area = 4 (area AOB)
$equals space 4 integral subscript 0 superscript 3 space straight y space dx space equals space 4 cross times 2 over 3 integral subscript 0 superscript 3 square root of 9 minus straight x squared end root space dx space equals space 8 over 3 integral subscript 0 superscript 3 square root of left parenthesis 3 right parenthesis squared minus straight x squared end root dx equals space 8 over 3 open square brackets fraction numerator straight x square root of left parenthesis 3 right parenthesis squared minus straight x squared end root over denominator 2 end fraction plus open parentheses 3 close parentheses squared over 2 sin to the power of negative 1 end exponent open parentheses straight x over 3 close parentheses close square brackets subscript 0 superscript 3 equals space 8 over 3 open square brackets open curly brackets fraction numerator 3 square root of 9 minus 9 end root over denominator 2 end fraction plus 9 over 2 sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis close curly brackets space minus space open curly brackets 0 plus 9 over 2 sin to the power of negative 1 end exponent left parenthesis 0 right parenthesis close curly brackets close square brackets equals space 8 over 3 open square brackets open curly brackets 0 plus 9 over 2 cross times straight pi over 2 close curly brackets minus open curly brackets 0 plus 9 over 2 cross times 0 close curly brackets close square brackets space equals space 8 over 3 cross times 9 over 2 cross times straight pi over 2 space equals space 6 space straight pi space sq. space units$

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Short Answer Type

20. Find the area of the region bounded by the ellipse

straight x squared over 16 plus straight y squared over 9 space equals 1.

151 Views