TBIL-Cal Polar Arclength (CO5)

Section 7.5 Polar Arclength (CO5)

Recall that the length of a parametric curve is given by

\int_{t = a}^{t = b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t .

Let

x (t) = r \cos (θ)

and

y (t) = r \sin (θ)

and show that the length of a polar curve

r = f (θ)

with

α \leq θ \leq β

is given by

\int_{θ = α}^{θ = β} \sqrt{{(r)}^{2} + {(\frac{d r}{d θ})}^{2}} d θ .

Find an integral computing the arclength of the polar curve defined by

r = 3 \cos (θ) - 2

π / 3 \leq θ \leq π .

Find the length of the cardioid

r = 1 - \cos (θ) .

Figure 178. Video for CO5

Calculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version