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Vector Algebra

Question
CBSEENMA12032765

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

Solution
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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