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Application of Integrals

Name: Zigya - For The Curious Learner
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Short Answer Type

61.
Find the area of the region {(x, y): x² + y² ≤ 1 ≤ x + y}.

The given region is
   $open curly brackets open parentheses straight x comma space straight y close parentheses colon straight x squared plus straight y squared less or equal than 1 less or equal than straight x plus straight y close curly brackets$
Consider the equations
   $straight x squared plus straight y squared space equals 1$ ...(1)
and x + y = 1 ...(2)
From (2), y = 1 - x ...(3)

Putting this value of y in (1), we get,
$straight x squared plus left parenthesis 1 minus straight x right parenthesis squared space equals space 1 space space space rightwards double arrow space space 2 straight x squared minus 2 straight x space equals space 0$
$rightwards double arrow$ $straight x squared minus straight x space equals 0 space space space rightwards double arrow space space straight x left parenthesis straight x minus 1 right parenthesis space equals space 0 space space space rightwards double arrow space space straight x space equals space 0 comma space space 1$
$therefore space space space from space left parenthesis 3 right parenthesis comma space space straight y space equals space 1 minus 0 comma space space 1 minus 1 space equals space 1 comma space space 0 therefore space space space circle space left parenthesis 1 right parenthesis space and space st. space line space straight x space plus space straight y space equals space 1 space space intersect space in space the space points space straight A thin space left parenthesis 1 comma space 0 right parenthesis space and space straight B left parenthesis 0 comma space 1 right parenthesis.$
Required area = Area of shaded region = $integral subscript 0 superscript 1 square root of 1 minus straight x squared end root space dx space space minus space integral subscript 0 superscript 1 left parenthesis 1 minus straight x right parenthesis space dx$
   $equals space open square brackets fraction numerator straight x square root of 1 minus straight x squared end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 space minus space open square brackets straight x minus straight x squared over 2 close square brackets subscript 0 superscript 1 equals space open square brackets open parentheses 0 plus 1 half sin to the power of negative 1 end exponent straight x close parentheses space open parentheses 0 plus 1 half sin to the power of negative 1 end exponent 0 close parentheses close square brackets space minus open square brackets open parentheses 1 minus 1 half close parentheses minus left parenthesis 0 minus 0 right parenthesis close square brackets equals space 1 half open parentheses straight pi over 2 close parentheses minus 1 half space equals open parentheses straight pi over 4 minus 1 half close parentheses space sq. space units. space$