Draw a graph of y2 = 16 x and x2 = 16y, and evaluate the area

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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

56.

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

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Long Answer Type

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

The equations of curves are

straight y squared space equals space 16 space straight x

...(1)
and

straight x squared space equals 16 space straight y

...(2)
From (2),

straight y space equals space straight x squared over 16

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

straight x to the power of 4 over 256 space equals space 16 space straight x space space space space or space space space straight x to the power of 4 space equals space 4096

or

straight x space left parenthesis straight x cubed minus 4096 right parenthesis space equals space 0

therefore space space space space space space space space space space space straight x space equals space 0 comma space space space 16

therefore space space space from space space left parenthesis 3 right parenthesis comma space space straight y space equals 0 comma space space 16 therefore space space space space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space in

O(0, 0), P(16, 16)
From P, draw PM

perpendicular space straight x minus axis.

Required area = Area OAPB = Area OBPM - area OAPM

equals space integral subscript 0 superscript 16 square root of 16 space straight x end root dx space space minus integral subscript 0 superscript 16 straight x squared over 16 dx space equals space 4 integral subscript 0 superscript 16 straight x to the power of 1 half end exponent dx space minus space 1 over 16 integral subscript 0 superscript 16 straight x squared space dx equals space 4 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 16 space minus space 1 over 16 open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 16 equals space 8 over 3 open square brackets left parenthesis 16 right parenthesis to the power of 3 over 2 end exponent minus 0 close square brackets minus 1 over 48 left square bracket left parenthesis 16 right parenthesis cubed minus 0 right square bracket equals space 8 over 3 cross times 64 minus 1 over 48 cross times 4096 space equals space 512 over 3 minus 256 over 3 space equals space 256 over 3 space sq. space units. space

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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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