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Vector Algebra

Question
CBSEENMA12032694

Find the area of the region bounded by the curve straight y space equals space 2 square root of 1 minus straight x squared end root and x-axis. Draw a rough sketch..

Solution
The given equations are
straight y space equals 2 square root of 1 minus straight x squared end root ...(1)
(In the first quadrant)
and y = 0    ...(2)
[∴ x-axis has equation y = 0]
From (1) and (2), we get,
                    0 space equals space 2 square root of 1 minus straight x squared end root space space space space space space space space rightwards double arrow space space space space space square root of 1 minus straight x squared end root space equals space 0
rightwards double arrow space space space 1 minus straight x squared space equals space 0 space space space space space space space space space space space space rightwards double arrow space space straight x squared space equals space 1 space space space space space space space space rightwards double arrow space space space space straight x space equals space minus 1 comma space space 1
Required area OAB = integral subscript 0 superscript 1 straight y space dx space equals space 2 integral subscript 0 superscript 1 square root of 1 minus straight x squared space end root space dx
                           equals space 2 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
equals space open square brackets straight x square root of 1 minus straight x squared end root plus sin to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1
equals left parenthesis 0 plus sin to the power of negative 1 end exponent 1 right parenthesis space minus space left parenthesis 0 plus sin to the power of negative 1 end exponent 0 right parenthesis space equals space 0 plus straight pi over 2 minus left parenthesis 0 plus 0 right parenthesis space equals space straight pi over 2

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