Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

Verified step by step guidance

1. Identify the function given as \(x = g(y)\) and the interval over which you want to find the curve length, from \(y = c\) to \(y = d\).
2. Recall the formula for the length of a curve expressed as \(x = g(y)\), which is given by the integral: \[L = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy\]
3. Compute the derivative \(\frac{dx}{dy}\) by differentiating \(g(y)\) with respect to \(y\).
4. Substitute \(\frac{dx}{dy}\) into the integral formula to get: \[L = \int_{c}^{d} \sqrt{1 + \left(g'(y)\right)^2} \, dy\]
5. Evaluate the integral over the interval \([c, d]\) to find the length of the curve.

Key Concepts

Arc Length Formula for Parametric Curves

The arc length of a curve defined by x = g(y) between y = c and y = d is found by integrating the square root of 1 plus the derivative of x with respect to y squared. This formula accounts for the infinitesimal distances along the curve, summing them to find the total length.

Derivative of the Function x = g(y)

To apply the arc length formula, you need the derivative dx/dy, which measures how x changes with respect to y. This derivative is essential because it determines the slope of the curve and influences the length calculation by affecting the integrand.

Definite Integration over the Interval [c, d]

After setting up the integrand involving the derivative, you compute the definite integral from y = c to y = d. This integration sums the infinitesimal arc lengths along the curve, yielding the total length between the specified bounds.