Calculate the length of the path over the given interval: (sin6t, cos6t), 0<t<pi
6pi
To calculate the length of a path over a given interval, we need to use the formula for arc length in parametric equations. The formula is:
L = ∫√[ (dx/dt)^2 + (dy/dt)^2 ] dt
where dx/dt and dy/dt are the derivatives of x and y with respect to t, and the integral is taken over the given interval.
In this case, the parametric equations are x = sin(6t) and y = cos(6t), and the interval is 0 < t < π.
First, let's find the derivatives:
dx/dt = 6cos(6t)
dy/dt = -6sin(6t)
Next, let's substitute these derivatives into the arc length formula:
L = ∫√[ (6cos(6t))^2 + (-6sin(6t))^2 ] dt
Simplifying the expression inside the square root:
L = ∫√[ 36cos^2(6t) + 36sin^2(6t) ] dt
L = ∫√[ 36(cos^2(6t) + sin^2(6t))] dt
L = ∫√[ 36 ] dt
L = ∫6 dt
Now, we integrate with respect to t:
L = 6t + C
Finally, we evaluate the integral over the given interval (0 < t < π):
L = 6π - 0 + C
L = 6π + C
So, the length of the path over the interval (0 < t < π) is 6π units.