Question

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).
c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.

Accepted Answer

Identify the regions Rₙ as the areas bounded between the nth and (n+1)st turns of the spiral given by $$r = e$$^{-	heta}$$, where $$	heta$$ ranges over the first and second quadrants, i.e., 0$$ \leq 	heta \leq \pi$$ for each turn. Express the area A_n$ of the region R_n$ using the formula for the area in polar coordinates between two curves or two values of $$	heta$$:

A_n$$ = \frac{1}{2} \int_{n\pi}^{(n+1)\pi} $$r^2$$ \, d	heta$$

Since r$$ = $$e^{-	heta}$$, substitute to get

$$A_n = \frac{1}{2} \int_{n\pi}^{(n+1)\pi} e$$^{-2	heta}$$ \, d	heta. $$Evaluate the integral for $$A_n$$:

$$A_n = \frac{1}{2} \left[ \frac{e$$^{-2	heta}$$}{-2} ight]_{n\pi}^{(n+1)\pi} = -\frac{1}{4} \left( e$$^{-2(n+1)\pi}$$ - e$$^{-2n\pi}$$ ight)$$

Rewrite this as

$$A_n = \frac{1}{4} \left( e$$^{-2n\pi}$$ - e$$^{-2(n+1)\pi}$$ ight). $$Write the expression for the ratio $$\frac{A_{n+1}}{A_n}$$:

$$\frac{A_{n+1}}{A_n} = \frac{e$$^{-2(n+1)\pi}$$ - e$$^{-2(n+2)\pi}$$}{e$$^{-2n\pi}$$ - e$$^{-2(n+1)\pi}$$}$$

Factor terms to simplify the ratio, for example, factor out $$e^{-2n\pi} in $$numerator and denominator. Take the limit as $$n 	o \infty of $$the ratio $$\frac{A_{n+1}}{A_n} by $$analyzing the behavior of the exponential terms. Since $$e^{-2n\pi}$$ tends to zero as $$n$$ grows large, simplify the expression accordingly to find the limit.

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).
c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates and Area Calculation

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Sequences and Ratio of Areas

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).c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates and Area Calculation

Behavior of Exponential Functions and Limits

Sequences and Ratio of Areas

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).
c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.