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Application of Integrals

Name: Zigya - For The Curious Learner
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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.

The equation of parabola is
y = x² ...(1)
The equation of line is
y = 4
This parabola is symmetrical about y-axis
$therefore space space space requird space area space POQRP space equals space 2 space left parenthesis area space OQRO right parenthesis$
$equals space 2 space integral subscript 0 superscript 4 straight x space dy space equals space 2 space integral subscript 0 superscript 4 square root of straight y space dy equals space 2 integral subscript 0 superscript 4 straight y to the power of 1 half end exponent dy space equals space 2 open square brackets fraction numerator straight y to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 4 space equals space 4 over 3 open square brackets straight y to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 4 space equals space 4 over 3 open square brackets 4 to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 open square brackets left parenthesis 2 squared right parenthesis to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 cross times 2 cubed space equals space 4 over 3 cross times 8 space equals space 32 over 3 space sq. space units$