00:01
The problem is find the exact length of the curve.
00:05
X is equal to 3 times cosine t minus cosine 3 t.
00:10
Y is equal to 3 times sine t minus sine 3 t and t is between 0 and pi.
00:18
So the length of the curve, l is equal to integral from 0 to pi.
00:24
Root of dx d t square plus the y d t square and d t so we can see d x d t is negative 3 times sine t plus 3 sine 3 t and the d y d t is 3 times cosine square and the d y d t is 3 times cosine t and minus 3 times cosine 3 t square and d t.
01:11
And this is equal to the integral from 0 to pi root of 9 sine t squared plus 9 3 t squared minus 18 times sine t sine 3 t plus 9 cosine square plus 9 cosine plus 9 cosine 3t square minus 18 times cosine t cosine 3 t and d t and here 9 sine 2 square plus 9 cosine square is 18 and 9 3d square plus uh sorry 9 sine t square plus 9 cosine t square is 9 and 9 sine 3d square plus 9 cosine 3t square is also 9 so this is equal to the integral from 0 to pi root of 18 minus 18 times sine t, sine 3t plus cosine t, cosine, cosine 3t.
03:00
Here, we use the formula cosine a minus b is equal to so, cosine a, cosine b, plus sine a, sine b...