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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.

The equation of parabola is y² = x
From the given condition, area OAP under y² = x between x = 0 and x = a = area ABQP under y² = x between x = a and x = 4.
$therefore space space space integral subscript 0 superscript straight a straight y space dx space equals space integral subscript straight a superscript 4 space straight y space dx rightwards double arrow space space integral subscript 0 superscript straight a square root of straight x space dx space equals space integral subscript straight a superscript 4 square root of straight x space dx rightwards double arrow space space integral subscript 0 superscript straight a straight x to the power of 1 half end exponent dx space equals space integral subscript straight a superscript 4 straight x to the power of 1 half end exponent dx rightwards double arrow space space 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 straight a space equals 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 straight a superscript 4 space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript straight a space equals space open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript straight a superscript 4 rightwards double arrow space space space space straight a to the power of 3 over 2 end exponent minus 0 space equals space 4 to the power of 3 over 2 end exponent minus straight a to the power of 3 over 2 end exponent space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space 2 space straight a to the power of 3 over 2 end exponent space equals space 8 space space space rightwards double arrow space space straight a to the power of 3 over 2 end exponent space equals space 4 space space space space rightwards double arrow space space straight a space space equals 4 to the power of 4 over 3 end exponent$