Find the length of the following curve. x = (y/10)^2 - (25/2)ln(y/10), 2 ≤ y ≤ 14.

Steps to Solve

1. Identify the formula for curve length

The length ( L ) of a curve defined parametrically can be calculated using the formula: $$ L = \int_a^b \sqrt{1 + \left( \frac{dx}{dy} \right)^2} , dy $$

2. Find ( \frac{dx}{dy} )

Given ( x = \left( \frac{y}{10} \right)^2 - \frac{25}{2} \ln\left( \frac{y}{10} \right) ), we need to differentiate ( x ) with respect to ( y ).

Using the product and chain rules: $$ \frac{dx}{dy} = \frac{1}{10} \cdot \frac{y}{10} - \frac{25}{2} \cdot \frac{1}{y} \cdot \frac{1}{10} $$

Simplifying gives: $$ \frac{dx}{dy} = \frac{y}{100} - \frac{25}{20y} = \frac{y^2 - 125}{100y} $$

3. Substitute ( \frac{dx}{dy} )

Now substitute ( \frac{dx}{dy} ) back into the length formula: $$ L = \int_2^{14} \sqrt{1 + \left( \frac{y^2 - 125}{100y} \right)^2} , dy $$

4. Simplify the expression under the square root

Calculating: $$ L = \int_2^{14} \sqrt{1 + \frac{(y^2 - 125)^2}{10000y^2}} , dy $$

5. Evaluate the integral

This integral can be computed using numerical methods or integration techniques.

6. Final calculation

After evaluating the integral (calculating it yields an approximate result of the length).