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Vector Algebra

Question
CBSEENMA12032813

Area lying between the curves y2 = 4x and y = 2x  is

  • 2 over 3
  • 1 third
  • 1 fourth
  • 3 over 4

Solution

B.

1 third

The equations of the curves are
                         straight y squared space equals space 4 straight x                         ...(1)
           and         straight y equals space 2 straight x                           ...(2)
From (1) and (2),
                          4 straight x squared space equals space 4 straight x
or               4 straight x left parenthesis straight x minus 1 right parenthesis space equals space 0.
therefore space space space space space space space space space straight x space equals space 0 comma space space 1
therefore space space space from space left parenthesis 2 right parenthesis comma space space straight y space equals space 0 comma space 2
therefore 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 space space straight A left parenthesis 6 comma space 2 right parenthesis

Required area  = integral subscript 0 superscript 1 straight y space dx space equals space integral subscript 0 superscript 1 2 space straight x space dx space equals space 2 space integral subscript 0 superscript 1 straight x to the power of 1 half end exponent dx minus 2 integral subscript 0 superscript 1 straight x space dx space equals space 2 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 1 space minus space 2 open square brackets straight x squared over 2 close square brackets subscript 0 superscript 1
                       equals space 4 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 1 space minus space open square brackets straight x squared close square brackets subscript 0 superscript 1 space equals space 4 over 3 left parenthesis 1 minus 0 right parenthesis minus space left parenthesis 1 minus 0 right parenthesis space equals space 4 over 3 minus 1 space equals space 1 third


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