Find the exact length of the curve. y = ln(1-x²), 0 ≤ x ≤ 1/7.

Solution:

Given, y = ln(1 -x² ), 0 ≤ x ≤ 1/7

Length of the curve, y = f(x) from x = a to x = b is given by:

\(\int_{a}^{b}\sqrt{1+f'(x)^{2}} .dx\)

y = ln(1-x²)

Differentiating w.r.t. x,

dy/dx = 1/(1 - x²) × (-2x) = -2x/(1 - x²)

Length of the curve,

= \(\int_{0}^{1/7}\sqrt{1+\left [\frac{-2x}{(1-x^{2})} \right ]^{2}} .dx\)\

= \(\int_{0}^{1/7}\sqrt{\left [ \frac{1 +x^{4}-2x^{2}+4x^{2}}{(1-x^{2})^{2}} \right ]} .dx\)

= \(\int_{0}^{1/7}\sqrt{\frac{(1+x^{2})^{2}}{(1-x^{2})^{2}}} . dx\)

= \(\int_{0}^{1/7}\frac{1+x^{2}}{1-x^{2}} . dx\)

{By dividing numerator by denominator and expressing using

Dividend/Divisor = Quotient + (Remainder/Divisor)]}

= \(\int_{0}^{1/7}-1 + \frac{2}{1-x^{2}}.dx\)

= \(-x \Biggr|_{0}^{1/7} + 2 \frac{1}{2} log\left | \frac{1 + x}{1-x} \right |\Biggr|_{0}^{1/7}\)

= \(-\left ( \frac{1}{7}-0 \right ) + log \left | \frac{1+ \frac{1}{7}}{1-\frac{1}{7}} \right |-log\left | \frac{1+0}{1-0} \right |\)

= \(-\frac{1}{7} + log\left | \frac{8}{6} \right |- log (1)\)

= \(-\frac{1}{7} + log\left | \frac{8}{6} \right |- 0\)

= log(4/3) - 1/7

Therefore, the exact length of the curve, y = ln(1- x²), 0 ≤ x ≤ 1/7 is log(4/3) - 1/7.

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Find the exact length of the curve. y = ln(1- x²), 0 ≤ x ≤ 1/7.

Summary:

The exact length of the curve, y = ln(1-x²), 0 ≤ x ≤ 1/7 is log(4/3) - 1/7.