91Ó°ÊÓ

Line integration is a powerful tool in calculus used to find the length of a curve or path traced out by a function. It is particularly useful when dealing with irregular shapes where straightforward measurement is difficult.
To compute the length of a line segment over a specific interval, we use the formula:

• \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]

In this expression, \( a \) and \( b \) represent the endpoints of the interval on the \( x \)-axis, and \( \frac{dy}{dx} \) is the derivative of the function. This formula accounts for the slope of the curve and the additional length added by any variation in \( y \).
By evaluating this integral, we sum up all infinitesimally small line segments along the curve to find the total length.

The distance formula is a practical way to calculate the straight-line distance between two points in a coordinate space. It stems from the Pythagorean theorem and is written as:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. This formula essentially determines the hypotenuse of a right triangle, where the differences in \( x \) and \( y \) coordinates form the two other sides.
Using the distance formula provides a straightforward way to verify the length found through more complex methods, such as line integration. It can simplify checks and balances in calculations, ensuring accuracy across different methods.

A definite integral represents the accumulation of quantities, which could be area under a curve, length of a curve, or even total distance traveled. When used to compute the length of a curve, it aims to total up all infinitesimal contributions over a specified interval.
In line integration, a definite integral helps sum small segments to find the complete length. For a function \( f(x) \), the definite integral \( \int_a^b f(x) \, dx \) is evaluated over the interval \([a, b]\).
The result is calculated by finding an antiderivative (if it exists) and subtracting the antiderivative's values at \( a \) and \( b \). In the exercise, the integral

• \[ \int_{1}^{3} \sqrt{5} \, dx = 2\sqrt{5} \]

showcases how definite integrals can help find a precise segment length, affirming the distance through mathematical verification.