Find the lengths of the curves in Exercises 1-12 . If you...

Find the lengths of the curves in Exercises $1-12 .$ If you have graphing
software, you may want to graph these curves to see what they look
like.
$$x=\left(y^{3 / 2} / 3\right)-y^{1 / 2} \text { from } y=1 \text { to } y=9$$

Key Concepts

Definite Integration

Definite integration is the process of evaluating the integral of a function between specified limits, which in this case correspond to the endpoints of the curve. Once the integrand involving the derivative is established, definite integration is used to sum the contributions of each differential arc length, resulting in the total length of the curve over the given interval.

Curve Analysis

Understanding the general behavior of curves, such as through graphing, is an important aspect of calculus. Graphing helps provide a visual confirmation of the curve's structure and continuity, ensuring that the chosen method and interval for arc length calculation are appropriate.

Arc Length Formula

The arc length formula in calculus is used to compute the length of a curve. For a curve described by a function x = f(y), the length between two points is calculated using the integral L = ? from a to b ?(1 + (dx/dy)²) dy. This formula arises from approximating the curve by infinitesimal line segments and summing their lengths, then taking the limit as the segment length approaches zero.

Differentiation

Differentiation is a fundamental concept in calculus that measures the rate of change of a function. In the context of finding curve lengths, computing the derivative of the function (here, dx/dy) is essential since it is used in the arc length integrand. The derivative quantifies the slope of the curve at any point, which directly influences the curve's length.