Find the area of the region bounded by y 2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant. - Wired Faculty

Vector Algebra

Question

Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

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Solution

The equation of curve is y² = 9x, which is right handed parabola.
Two lines are x = 2, x = 4.
Required area = Area ABCD

equals space integral subscript 2 superscript 4 straight y space dx space equals space integral subscript 2 superscript 4 3 square root of straight x space dx space equals space 3 space integral subscript 2 superscript 4 straight x to the power of 1 half end exponent dx space equals space 3 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 2 superscript 4 space equals space 3 cross times 2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 2 superscript 4 equals space 2. space open square brackets left parenthesis 4 right parenthesis to the power of 3 over 2 end exponent minus left parenthesis 2 right parenthesis to the power of 3 over 2 end exponent close square brackets space equals space 2 space left square bracket space 8 space minus square root of 8 right square bracket space equals space 2 left parenthesis 8 minus 2 square root of 2 right parenthesis equals space left parenthesis 16 minus 4 square root of 2 right parenthesis space sq. space units.

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