Question

Can someone explain in details please!! 1. The starship Enterprise is taking evasive maneuvers. Its position can be modeled by the equations x(t) = t + cos(t) y(t) = t sin(t): where x and y are measured in km. (We assume that the ship doesn't vary its z-coordinate.) Graph the path of the ship in the xy-plane from t = 0 to t = 10 seconds. Use Simpson's rule with n = 6 to approximate the distance traveled by the ship from t = 0 to t = 10. Compute a better approximation using your calculator or Wolfram. In particular your response should address the following questions.

Answer 1

_ What is the formula for the arc length of a parametric curve?
_ Where does the formula for arc length come from?
_ What is the idea behind Simpson's Rule. In particular you need to find a way to explain Simpson's rule without getting bogged down with lots of mundane calculations.
_ How does your answer compare to the one produced by the computer.

Answer 2

I can get you started, you are given parametric position equations for the position vector, looks like:
\[r = <x(t), y(t)>=<t + cos(t), tsin(t)> \]

x(t) tells you where the x position of the ship is at time t, and y(t) tells you were its y position is at time t, resulting in an ordered pair (x(t), y(t)).

If you have to graph this by hand, find a few points and plug it in, but I'd recommend some utility if you are able to.

As far the arc length, image to come: