21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 
y = ln x, for 1≤x≤4

Verified step by step guidance

1. Recall the formula for the arc length of a curve given by a function \( y = f(x) \) on the interval \( [a, b] \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
2. Identify the function and the interval: here, \( y = \ln x \) and the interval is \( 1 \leq x \leq 4 \).
3. Compute the derivative \( \frac{dy}{dx} \) of \( y = \ln x \): \[ \frac{dy}{dx} = \frac{1}{x} \]
4. Substitute \( \frac{dy}{dx} \) into the arc length formula: \[ L = \int_1^4 \sqrt{1 + \left(\frac{1}{x}\right)^2} \, dx = \int_1^4 \sqrt{1 + \frac{1}{x^2}} \, dx \]
5. Simplify the expression under the square root: \[ \sqrt{1 + \frac{1}{x^2}} = \sqrt{\frac{x^2 + 1}{x^2}} = \frac{\sqrt{x^2 + 1}}{x} \] So the integral becomes: \[ L = \int_1^4 \frac{\sqrt{x^2 + 1}}{x} \, dx \]

Key Concepts

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral ∫ from a to b of √(1 + (dy/dx)²) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.

Derivative of the Natural Logarithm Function

For y = ln(x), the derivative dy/dx is 1/x. Understanding this derivative is essential to substitute into the arc length formula, as it determines the slope of the curve at each point.

Simplifying the Integral Expression

After substituting dy/dx into the arc length formula, the integral often requires algebraic simplification before evaluation. Simplifying √(1 + (1/x)²) to √(1 + 1/x²) = √((x² + 1)/x²) = √(x² + 1)/x helps in setting up the integral for numerical or calculator-based evaluation.