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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.

Name: Zigya - For The Curious Learner
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The equation of parabola is
y² = 4 x
The equation of line is
x = 3
Also, we know that parabola is symmetric about x-axis
∴ required area = 2 (area ORPO)

$equals space 2 integral subscript 0 superscript 3 2 square root of straight x space dx space equals space 4 space integral subscript 0 superscript 3 straight x to the power of 1 half end exponent dx equals space 4 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 3 space equals space 8 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 3 space equals space 8 over 3 open parentheses 3 to the power of 3 over 2 end exponent minus 0 close parentheses space equals space 8 over 3 cross times square root of 27 equals space 8 over 3 cross times 3 square root of 3 space equals space 8 square root of 3 space sq. space units.$