Final answer:
To find the arc length of the curve y = x⁴ / 4 + 1 / (8x²) on the interval [1,4], we can use the formula for arc length: L = ∫sqrt(1 + (dy/dx)²) dx. This involves taking the derivative of y with respect to x, substituting it into the arc length formula, and evaluating the resulting integral.
Step-by-step explanation:
To find the arc length of the curve y = x⁴ / 4 + 1 / (8x²) on the interval [1,4], we can use the formula for arc length:
L = ∫sqrt(1 + (dy/dx)²) dx
In this case, dy/dx is the derivative of y with respect to x. Taking the derivative, we get:
dy/dx = x³ - (1 / (4x³))
Substituting this into the arc length formula:
L = ∫sqrt(1 + (x³ - (1 / (4x³)))²) dx
This integral can be evaluated using integration techniques such as substitution or integration by parts.