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).
a. Find the area Aₙ of Rₙ.

Accepted Answer

Identify the spiral given by the polar equation $$r = e$$^{-	heta}$$, where $$	heta \geq 0$$. The regions R_n$ are bounded between the n$th and (n+1$$)$$st turns of the spiral, which correspond to $$	heta = n\pi$$ and $$	heta = (n+1)\pi$$ respectively, since one full turn corresponds to an increase of $$\pi$$ in $$	heta$$ (covering the first and second quadrants). Recall the formula for the area of a region bounded by a polar curve between angles $$\alpha$$ and $$\beta$$:

A$$ = \frac{1}{2} \int_{\alpha}^{\beta} $$r^2$$ \, d	heta$$ Apply this formula to find the area A_n$ of the region R_n$ by integrating r^2$ = (e$$^{-	heta}$$)^2$$ = $$e^{-2	heta}$$ from $$	heta = n\pi to$$ $$	heta = (n+1)\pi$$:

$$A_n = \frac{1}{2} \int_{n\pi}^{(n+1)\pi} e$$^{-2	heta}$$ \, d	heta$$ Evaluate the integral $$\int e$$^{-2	heta}$$ \, d	heta. $$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} $$e^{-2	heta}$$ plus a constant of integration. 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 $$nth $$and $$(n+1)$st turns of the spiral.

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 video answer for a similar problem:

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Understanding Regions Bounded by Spiral Turns

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 video answer for a similar problem:

Key Concepts

Polar Coordinates and Curves

Area in Polar Coordinates

Understanding Regions Bounded by Spiral Turns

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.