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Vector Algebra

Question
CBSEENMA12032758

Find the area of the region bounded by two parabola 4 y = x2 and 4 x = y2

Solution

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

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