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## Q. 5.3

Calculate the arc length of the cardioid r = a(1 + cos θ), a > 0.

## Verified Solution

The curve is sketched in Figure 3. Since the curve is symmetric about
the polar axis, we calculate the arc length between θ =
0 and θ = π and then multiply
the result by
2. We have

\begin{aligned}\frac{s}{2} &=\int_{0}^{\pi} \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta=\int_{0}^{\pi} a \sqrt{(1+\cos \theta)^{2}+\sin ^{2}} d \theta \\&=a \int_{0}^{\pi} \sqrt{1+2 \cos \theta+\cos ^{2} \theta+\sin ^{2} \theta} d \theta \\&=a \int_{0}^{\pi} \sqrt{(2+2 \cos \theta)} d \theta=a \int_{0}^{\pi} \sqrt{\frac{4+4 \cos \theta}{2}} d \theta \\&=2 a \int_{0}^{\pi} \sqrt{\frac{1+\cos \theta}{2}} d \theta=2 a \int_{0}^{\pi} \cos \frac{\theta}{2} d \theta=\left.4 a \sin \frac{\theta}{2}\right|_{0} ^{\pi}=4 a .\end{aligned}

The total arc length is therefore 8a. 