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Vector Algebra

Question
CBSEENMA12032687
Wired Faculty App

Find the area of the region bounded by y2 = x - 2, x = 4, x = 6 and the x-axis in the first quadrant.

Solution

The equation of curve is y2 = x - 2. which is right handed parabola with vertex at (2, 0).
Two lines are x = 4 and x = 6
Required area  = Area ABCD
                         equals space integral subscript 4 superscript 6 straight y space dx space equals space integral subscript 4 superscript 6 square root of straight x minus 2 end root dx equals space integral subscript 4 superscript 6 left parenthesis straight x minus 2 right parenthesis to the power of 1 half end exponent dx equals space open square brackets fraction numerator left parenthesis straight x minus 2 right parenthesis 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 4 superscript 6 space equals space 2 over 3 open square brackets left parenthesis straight x minus 2 right parenthesis to the power of 3 over 2 end exponent close square brackets subscript 4 superscript 6 equals space 2 over 3 open square brackets 4 to the power of 3 over 2 end exponent minus 2 to the power of 3 over 2 end exponent close square brackets space equals space space 2 over 3 left parenthesis 8 minus 2 square root of 2 right parenthesis space sq. space units


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