Question

Consider the curve defined by y = (x⁶)/8 + (1/(12x⁴)) from x = 1 to x = 4. The length of this curve is L = ∫[1 to 4] √(1 + (f'(x))²) dx, where f'(x) is the derivative of y with respect to x.

Answer 1

To find the curve's length defined by y = (x⁶)/8 + (1/(12x⁴)), we need to calculate its derivative, square it, add 1, and evaluate the definite integral of the square root of this sum from x = 1 to x = 4.

Step-by-step explanation:

The student's question relates to finding the length of a curve defined by y = (x⁶)/8 + (1/(12x⁴)) between x = 1 and x = 4. This involves calculating the integral of the square root of 1 plus the derivative of the function squared, indicated as L = ∫[1 to 4] √(1 + (f'(x))²) dx. To proceed, we need to find the derivative of the function, f'(x), then substitute it into the integrand to evaluate the arc length.

Firstly, we compute f'(x) as follows:

• Take the derivative of the function y with respect to x.
• Simplify and square the derivative.
• Add 1 to the squared derivative as per the formula for the arc length.
• Take the square root of the entire expression to get the integrand.
• Finally, integrate this expression from x = 1 to x = 4 to find the arc length L.

This process will yield the exact length of the curve within the given bounds.