Spiral arc length Consider the spiral r=4θ, for θ≥0.

a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.

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Recall the formula for the length of a curve given in polar coordinates: \(L = \int_{\alpha}^{\beta} \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta\).

Identify the given function: \(r(\theta) = 4\theta\), and the interval for \(\theta\) is from \(0\) to \(\sqrt{8}\).

Compute the derivative of \(r(\theta)\) with respect to \(\theta\): \(\frac{dr}{d\theta} = 4\).

Substitute \(r(\theta)\) and \(\frac{dr}{d\theta}\) into the arc length formula to get: \(L = \int_0^{\sqrt{8}} \sqrt{(4\theta)^2 + 4^2} \, d\theta = \int_0^{\sqrt{8}} \sqrt{16\theta^2 + 16} \, d\theta\).

Factor out the constant inside the square root and prepare for a trigonometric substitution: \(L = \int_0^{\sqrt{8}} 4 \sqrt{\theta^2 + 1} \, d\theta\). Use the substitution \(\theta = \sinh u\) or \(\theta = \tan u\) to simplify the integral.

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Polar Coordinates and Parametric Curves

In polar coordinates, a curve is described by r as a function of θ. Understanding how to express the curve in terms of r(θ) and θ is essential, as it allows conversion to parametric form for length calculations using x = r cos θ and y = r sin θ.

Arc Length Formula for Polar Curves

The length of a curve defined in polar form r(θ) from θ = a to θ = b is given by the integral ∫ from a to b of √(r(θ)² + (dr/dθ)²) dθ. This formula combines the radius and its rate of change to measure the curve's length accurately.

Trigonometric Substitution in Integration

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions. By substituting variables with trigonometric functions, the integral becomes easier to evaluate, which is useful when finding the arc length of curves like spirals.

Spiral arc length Consider the spiral r=4θ, for θ≥0. a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.