Give the rough sketch of the curve y 2 = x and the line x = 4 and find the area between the curve and the line. - Wired Faculty

Vector Algebra

Question

Give the rough sketch of the curve y² = x and the line x = 4 and find the area between the curve and the line.

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Solution

The equation of parabola is
y² = x ...(1)
The equation of line is x = 4
Also, we know that parabola is symmetric about x-axis

therefore space space space required space area space equals space 2 space left parenthesis area space ORP right parenthesis

equals space 2 integral subscript 0 superscript 4 space straight y space dx space equals space 2 space integral subscript 0 superscript 4 square root of straight x space dx space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

equals space 2 space integral subscript 0 superscript 4 straight x to the power of 1 half end exponent 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 4 space 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 4 space equals space 4 over 3 open square brackets 4 to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 cross times square root of 64 space equals space 4 over 3 cross times 8 space equals space 32 over 3 space sq space. units.

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