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Application of Integrals

Name: Zigya - For The Curious Learner
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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

The equations of curves are
$straight y squared space equals space 4 straight x$ ...(1)
and $straight x squared space equals space 4 straight y$ ...(2)
From (2), $straight y space equals space straight x squared over 4$ ...(3)
Putting this value of y in (1),
$straight x to the power of 4 over 16 space equals space 4 space straight x space space space space or space space space straight x to the power of 4 space equals space 64 space straight x$
or $straight x left parenthesis straight x cubed minus 64 right parenthesis space equals space 0$
$therefore space space space straight x space equals space 0 comma space space 4 therefore space space space from space left parenthesis 3 right parenthesis comma space space space straight y space equals space 0 comma space space 4 because space space space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space in space straight O left parenthesis 0 comma space 0 right parenthesis comma space straight P space left parenthesis 4 comma space 4 right parenthesis.$

From P, draw PM $perpendicular$ x-axis.
Required area = Area OAPB
= Area OBPM - area OAPM
$equals space integral subscript 0 superscript 4 4 square root of straight x space dx space space minus integral subscript 0 superscript 4 straight x squared over 4 dx space equals space 2 integral subscript 0 superscript 4 straight x to the power of 1 half end exponent dx minus 1 fourth integral subscript 0 superscript 4 straight x squared space dx space equals space 2 space open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 4 space minus space 1 fourth open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 4 equals space 2 space cross times 2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 4 space minus space 1 fourth cross times 1 third open square brackets straight x cubed close square brackets subscript 0 superscript 4 space equals space 4 over 3 open square brackets left parenthesis 4 right parenthesis to the power of 3 over 2 end exponent minus 0 close square brackets minus 1 over 12 left square bracket left parenthesis 4 right parenthesis cubed minus 0 right square bracket equals space 4 over 3 cross times 8 minus 1 over 12 cross times 64 space equals space 32 over 3 minus 16 over 3 equals space 16 over 3 space sq. space units. space$