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Application of Integrals

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Long Answer Type

51. Find the area of the region enclosed between the two circles x² + y² = 4 and (x - 2)² + y² = 4.

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52. Find the area of the region enclosed between the two circles x² + y² = 1 and (x - 1)² + y² = 1.

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53. Find the area of the region enclosed between the two circles x² + y² = 1 and

open parentheses straight x minus 1 half close parentheses squared plus straight y squared space equals space 1.

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Short Answer Type

54. Find the area of the region bounded by two parabola 4 y = x² and 4 x = y²

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Long Answer Type

55. Find the area included between the two curves y² = 9x and ,x² = 9y. Also draw the rough sketch.

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Short Answer Type

56.

Calculate the area of the region bounded by the y = x² and x = y².

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Long Answer Type

57. Draw a graph of y² = 16 x and x² = 16y, and evaluate the area between them.

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58. Find the area of the region included between the parabolas y² = 4 a x and x² = 4 a y, a > 0.

The equations of curves are
$straight y squared space equals space 4 space straight a space straight x$ ...(1)
and $straight x squared space equals 4 space straight a space straight y$ ...(2)
From (2), $straight y equals fraction numerator straight x squared over denominator 4 space straight a end fraction$ ...(3)
Putting this value of y in (1),
$fraction numerator straight x to the power of 4 over denominator 16 space straight a squared end fraction space equals space 4 space straight a space straight x comma space space or space space space straight x to the power of 4 space equals space 64 space straight a cubed space straight x$
or $straight x left parenthesis straight x cubed minus 64 straight a cubed right parenthesis space equals 0$
∴ x = 0, 4 a
∴ from (3), y = 0, 4 a
∴ curves (1) and (2) intersect in O (0, 0), P (4 a, 4 a).
From P. draw PM ⊥ x-axis.
Required area = Area OAPB
= Area OBPM - area OAPM
$equals space integral subscript straight a superscript 4 straight a end superscript square root of 4 space straight a space straight x end root space dx space minus space integral subscript 0 superscript 4 straight a end superscript fraction numerator straight x squared over denominator 4 straight a end fraction dx space equals space 2 square root of straight a integral subscript 0 superscript 4 space straight a space end superscript straight x to the power of 1 half end exponent dx space minus space fraction numerator 1 over denominator 4 space straight a end fraction integral subscript 0 superscript 4 space straight a end superscript straight x squared space dx$
$equals space 2 square root of straight a open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 4 straight a end superscript space minus space fraction numerator 1 over denominator 4 straight a end fraction open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 4 straight a end superscript equals space 2 square root of straight a cross times 2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 4 straight a end superscript space minus space fraction numerator 1 over denominator space 4 straight a end fraction cross times 1 third open square brackets straight x cubed close square brackets subscript 0 superscript 4 straight a end superscript equals space fraction numerator 4 square root of straight a over denominator 3 end fraction open square brackets left parenthesis 4 space straight a right parenthesis to the power of 3 over 2 end exponent minus 0 close square brackets space minus space fraction numerator 1 over denominator 12 space straight a end fraction open square brackets left parenthesis 4 space straight a right parenthesis cubed space minus space 0 close square brackets equals space fraction numerator 4 square root of straight a over denominator 3 end fraction cross times 8 straight a to the power of 3 over 2 end exponent minus fraction numerator 1 over denominator 12 space straight a end fraction cross times 64 space straight a cubed space equals space 32 over 3 straight a squared minus 16 over 3 straight a squared space equals space 16 over 3 straight a squared space sq. space units.$