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 = 1/x, for 1 ≤ x ≤ 10

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 interval: here, \( y = \frac{1}{x} \) and \( x \) ranges from 1 to 10.
3. Compute the derivative \( \frac{dy}{dx} \) of \( y = \frac{1}{x} \). Use the power rule or rewrite \( y = x^{-1} \) and differentiate.
4. Square the derivative \( \left(\frac{dy}{dx}\right)^2 \) and add 1 inside the square root to form the integrand: \[ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
5. Write the integral for the arc length explicitly with the limits 1 to 10: \[ L = \int_1^{10} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] This integral can then be evaluated using a calculator or numerical methods.

Key Concepts

Arc Length Formula

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

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = 1/x. The derivative measures the slope of the curve at each point and is essential for computing the integrand √(1 + (dy/dx)^2).

Definite Integral Evaluation

After setting up the integral for arc length, evaluating it over the interval [1, 10] gives the total length. This may require simplification or numerical methods, such as using a calculator, especially when the integral cannot be expressed in elementary functions.