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Vector Algebra

Question
CBSEENMA12032760

Find the area included between the two curves y2 = 9x and ,x2 = 9y. Also draw the rough sketch.

Solution

The equations of curves are
                           straight y squared space equals 9 straight x                                             ...(1)
    and                 straight x squared space equals space 9 straight y                                            ...(2)
    From (1) and (2), we get
                open parentheses straight x squared over 9 close parentheses squared space equals space 9 space straight x space space space space space space space space or space space space space space space straight x to the power of 4 over 81 space equals space 9 straight x
therefore space space space straight x to the power of 4 minus 729 straight x space equals space 0 space space space space space space space space space rightwards double arrow space space straight x left parenthesis straight x cubed minus 729 right parenthesis space equals space 0
therefore space space straight x space equals space 0 comma space space 9
therefore space space straight y space equals space 0 comma space 9

therefore   curves (1) and (2) intersect in O(0, 0), P(9, 9)
      From P, draw PM space perpendicular x-axis. 
      Required area = Area OAPBO = Area OBPM - area OAPM
                              equals space integral subscript 0 superscript 9 square root of 9 straight x end root space dx space minus space integral subscript 0 superscript 9 straight x squared over 9 dx space equals space 3 integral subscript 0 superscript 9 straight x to the power of 1 half end exponent dx minus 1 over 9 integral subscript 0 superscript 9 straight x squared dx
    equals space 3 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 9 space minus space 1 over 9 open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 9 space equals space 2 space open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 9 space minus space 1 over 27 open square brackets straight x cubed close square brackets subscript 0 superscript 9
equals space 2 space open square brackets 9 to the power of 3 over 2 end exponent minus 0 close square brackets space minus space 1 over 27 open square brackets 9 cubed minus 0 close square brackets
equals space 2 space left parenthesis 27 minus 0 right parenthesis space minus space 1 over 27 left parenthesis 729 minus 0 right parenthesis
equals space 54 minus 27 space equals space 27 space sq. space units. space

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