# How do you find the distance travelled from t=0 to #t=pi# by an object whose motion is #x=3cos2t, y=3sin2t#?

##### 1 Answer

Oct 5, 2017

#### Explanation:

We seek the the distance travelled from

**Short Solution**

We note that the parametric equations are those of a circle of radius

Thus the distance travelled is the circumference of said circle:

# L = (2)(pi)(3) = 6 pi#

**Long Solution**

We can calculate the parametric arc length using,

# L = int_alpha^beta sqrt( dot(x)^2 + dot(y)^2 ) \ dt #

# \ \ = int_0^(pi) sqrt( (-6sin2t)^2 + (6cos2ty)^2 ) \ dt #

# \ \ = int_0^(pi) sqrt( 6^2(sin^2t+cos^2t) ) \ dt #

# \ \ = int_0^(pi) sqrt( 6^2 ) \ dt #

# \ \ = int_0^(pi) 6 dt #

# \ \ = [6 t]_0^(pi) #

# \ \ = 6pi #