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. Step 1: Recall the formula for the length of a curve. The length of a curve defined as x = g(y) between y = c and y = d is given by the integral: ${\int }_c^d\sqrt{1+\left(\frac{dx}{dy}\right)2^{}}dy$.
2. Step 2: Compute the derivative dx/dy. To find the derivative of x with respect to y, differentiate the function x = g(y) with respect to y. This will give you $\frac{dx}{dy}$.
3. Step 3: Square the derivative dx/dy. Once you have computed $\frac{dx}{dy}$, square it to obtain ${\left(\frac{dx}{dy}\right)}^2$.
4. Step 4: Substitute into the formula. Replace the squared derivative and the constant 1 into the square root part of the formula: $\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}$.
5. Step 5: Evaluate the integral. Integrate the expression $\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}$ with respect to y from y = c to y = d to find the length of the curve.

Key Concepts

Parametric Representation of Curves

In calculus, curves can be represented parametrically, where one variable is expressed in terms of another. For the curve defined by x = g(y), y serves as the parameter. This representation allows us to analyze the curve's properties, such as its length, by integrating with respect to the parameter.

Arc Length Formula

The arc length of a curve defined by a function can be calculated using the formula L = ∫√(1 + (dx/dy)²) dy. Here, dx/dy is the derivative of x with respect to y, which measures the rate of change of x as y varies. This formula accounts for the infinitesimal changes in both x and y, providing the total length of the curve between specified limits.

Definite Integrals

Definite integrals are used to calculate the total accumulation of a quantity over an interval. In the context of finding the length of a curve, the limits of integration (c and d) define the segment of the curve being measured. Evaluating the definite integral gives the exact length of the curve between these two points, incorporating the contributions of all infinitesimal segments.