9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = y⁴/4 + 1/8y², for 1≤y≤2

Verified step by step guidance

1. Step 1: Recall the formula for arc length of a curve given in parametric form: $L\; =\; \int$_a^b √(1 + (dx/dy)²) dy. Here, the curve is given as $x\; =\; y4/4\; +\; 1/(8y2)$, and the interval is $1\; \le \; y\; \le \; 2$.
2. Step 2: Compute the derivative $dx/dy$. Differentiate $x\; =\; y4/4\; +\; 1/(8y2)$ with respect to $y$. Use the power rule and chain rule: $dx/dy\; =\; y3-\; 1/(4y3)$.
3. Step 3: Square the derivative $dx/dy$. This gives $(dx/dy)2=\; (y3-\; 1/(4y3))2$. Expand the square if necessary for simplification.
4. Step 4: Substitute $(dx/dy)2$ into the arc length formula. The integrand becomes $\surd (1\; +\; (y3-\; 1/(4y3))2)$. Set up the integral: $L\; =\; \int$₁² √(1 + (y³ - 1/(4y³))²) dy.
5. Step 5: Evaluate the integral. Depending on the complexity of the integrand, you may need numerical methods or advanced techniques to compute the arc length. Ensure proper limits of integration are applied from $y\; =\; 1$ to $y\; =\; 2$.

Key Concepts

Arc Length Formula

The arc length of a curve defined by a function can be calculated using the formula L = ∫√(1 + (dy/dx)²) dy, where dy/dx is the derivative of the function with respect to y. This formula allows us to find the length of the curve over a specified interval by integrating the square root of one plus the square of the derivative.

Implicit Differentiation

When dealing with curves defined in terms of y, implicit differentiation is often used to find dy/dx. This technique involves differentiating both sides of an equation with respect to y, treating x as a function of y, which is essential for applying the arc length formula correctly.

Definite Integrals

Definite integrals are used to calculate the total length of the curve over a specific interval. In this context, the limits of integration correspond to the values of y (from 1 to 2) for which the arc length is being calculated, allowing us to evaluate the integral to find the exact length of the curve.