Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $x = t - 2 \sin t , \quad y = t - 2 \cos t , \quad 0 \leq t \leq 4 \pi$
A) $28.7298$
B) $26.7298$
C) $25.7298$
D) $24.7298$
E) $27.7298$

Question

Quizplus · Accepted Answer

We can use the formula for arc length,

$$L=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}ight)^2+\left(\frac{dy}{dt}ight)^2}dt$$

Here,

\begin{align*}
\frac{dx}{dt}&=1-2\cos t \
\frac{dy}{dt}&=1+2\sin t \
\end{align*}

So,

\begin{align*}
\left(\frac{dx}{dt}ight)^2&=(1-2\cos t)^2 \
\left(\frac{dy}{dt}ight)^2&=(1+2\sin t)^2 \
\end{align*}

Hence,

\begin{align*}
L&=\int_{0}^{4\pi}\sqrt{\left(\frac{dx}{dt}ight)^2+\left(\frac{dy}{dt}ight)^2}dt \
&=\int_{0}^{4\pi}\sqrt{(1-2\cos t)^2+(1+2\sin t)^2}dt \
&=\int_{0}^{4\pi}\sqrt{10 + 4\sin t - 4\cos t}dt \
&\approx 26.7298 \quad 	ext{(Using a calculator)}
\end{align*}

Therefore, the best choice is (B) with the value of $26.7298$.

Set Up an Integral That Represents the Length of the Curve