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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.

Name: Zigya - For The Curious Learner
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The equation of parabola is
y² = x ...(1)
The equation of line is
x = 3
Also. we know that parabola is symmetric about x-axis.
∴ required area = 2 (area ORP)
$equals space 2 integral subscript 0 superscript 3 straight y space dx space equals space 2 space integral subscript 0 superscript 3 space square root of straight x space dx equals space 2 space integral subscript 0 superscript 3 straight x to the power of 1 half end exponent dx space equals space 2 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 3 space equals space 4 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 3 equals space 4 over 3 open square brackets 3 to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 cross times square root of 27 space equals space 4 over 3 cross times 3 square root of 3 space equals space 4 square root of 3 space sq. space units.$