Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).
a. Find the area Aₙ of Rₙ.

Verified step by step guidance

1. Identify the spiral given by the polar equation \(r = e^{-\theta}\), where \(\theta \geq 0\). The regions \(R_n\) are bounded between the \(n\)th and \((n+1)\)st turns of the spiral, which correspond to \(\theta = n\pi\) and \(\theta = (n+1)\pi\) respectively, since one full turn corresponds to an increase of \(\pi\) in \(\theta\) (covering the first and second quadrants).
2. Recall the formula for the area of a region bounded by a polar curve between angles \(\alpha\) and \(\beta\): \(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\)
3. Apply this formula to find the area \(A_n\) of the region \(R_n\) by integrating \(r^2 = (e^{-\theta})^2 = e^{-2\theta}\) from \(\theta = n\pi\) to \(\theta = (n+1)\pi\): \(A_n = \frac{1}{2} \int_{n\pi}^{(n+1)\pi} e^{-2\theta} \, d\theta\)
4. Evaluate the integral \(\int e^{-2\theta} \, d\theta\). Use the substitution method or recall that the integral of \(e^{ax}\) is \(\frac{1}{a} e^{ax}\), so here it becomes \(-\frac{1}{2} e^{-2\theta}\) plus a constant of integration.
5. Substitute the limits \(n\pi\) and \((n+1)\pi\) into the evaluated integral and multiply by \(\frac{1}{2}\) to express \(A_n\) in terms of \(n\). This will give the exact area of the region \(R_n\) bounded by the \(n\)th and \((n+1)\)st turns of the spiral.

Key Concepts

Polar Coordinates and Curves

Polar coordinates represent points using a radius and angle (r, θ), which is essential for describing curves like spirals. The given spiral r = e^(-θ) is defined in terms of θ, making it natural to analyze areas bounded by turns using polar integration.

Area in Polar Coordinates

The area enclosed by a curve in polar coordinates between angles θ = a and θ = b is found using the integral A = 1/2 ∫[a to b] (r(θ))^2 dθ. This formula is crucial for calculating the area of regions bounded by consecutive turns of the spiral.

Understanding Regions Bounded by Spiral Turns

The region Rₙ lies between the nth and (n+1)st turns of the spiral, corresponding to θ values nπ/2 and (n+1)π/2 in the first and second quadrants. Identifying these angular bounds is key to setting up the integral for the area Aₙ.