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

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Application of Integrals

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

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56.
Calculate the area of the region bounded by the y = x² and x = y².

The equations of parabolas are

straight y squared space equals space straight x

...(1)
and

straight x squared space equals space straight y

...(2)
From (2),

straight y equals straight x squared

....(3)
Putting this values of y in (1), we get,

straight x to the power of 4 space equals space straight x space space space or space space space straight x left parenthesis straight x cubed minus 1 right parenthesis space equals space 0

rightwards double arrow space space space straight x space equals space 0 comma space 1

therefore space space space space from space left parenthesis 3 right parenthesis comma space space straight y space equals space 0 comma space space space 1 therefore space space space space parabolas space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space in space straight O left parenthesis 0 comma space 0 right parenthesis comma space straight P left parenthesis 1 comma space 1 right parenthesis.

From P draw PM

perpendicular

x-axis
Required area = Area of region OAPB = Area of region OBPM - area of region OAPM

equals integral subscript 0 superscript 1 square root of straight x space dx space minus space integral subscript 0 superscript 1 straight x squared dx

equals space open square brackets fraction numerator straight x to the power of 3 divided by 2 end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 1 space minus space open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 1 space equals space 2 over 3 open square brackets straight x to the power of 3 divided by 2 end exponent close square brackets subscript 0 superscript 1 space minus space 1 third open square brackets straight x cubed close square brackets subscript 0 superscript 1 equals space 2 over 3 left square bracket 1 minus 0 right square bracket space minus 1 third left square bracket 1 minus 0 right square bracket space equals space 2 over 3 minus 1 third space equals space 1 third space sq. space units.

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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.

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59. Draw the rough sketch of y² = x + 1 and y² = - x + 1 and determine the area enclosed by the two curves.

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60. Find the area of the region
{(x, y): 0 ≤ y ≤ x² + 1 , 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2}.

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