CPM Homework Help : APCALC Problem 12-14

Multiple Choice: The arc length of the cardioid given by $x(t) = 2\cos(t) - \cos(2t)$ and $y = 2\sin(t) - \sin(2t)$ over the interval $0 ≤ t ≤ 2π$ can be calculated by evaluating the integral:

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 - 8 \operatorname { cos } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 + 8 \operatorname { cos } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 - 8 \operatorname { sin } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 8 + 8 \operatorname { sin } ( t ) } d t$

$\int _ { 0 } ^ { 2 \pi } \sqrt { 5 - 4 \operatorname { cos } ( t ) } d t$

Hint:

$\text{arc length } =\int_0^{2\pi}\sqrt{(x^\prime(t))^2+(y^\prime(2))^2}dt$

Step:

$x^\prime(t)=-2\sin(t)+2\sin(2t)$

$y^\prime(t)=2\cos(t)-2\cos(2t)$