Originally Posted by

**harold** This one has been giving me a hard time:

Consider the parametric curve $\displaystyle c(t) = (t cos(t), t sin(t)), 0 \le t \le 2\pi$ (a spiral). Recall from calculus that the arclength formula of a parametric curve $\displaystyle c(t) = (x(t), y(t))$ with $\displaystyle a\le t\le b$ is given by $\displaystyle l(c)=\int_a^b\sqrt{x'(t)^2 + y'(t)^2} \, dt$.

What I need to figure out is:

1) How to calculate the arclength of the spiral by hand.

2) Use Simpson's Rule to approximate the integral. Find the error bound using the appropriate error formula, and compare it to the actual error.

3) Determine the number of subintervals $\displaystyle n$ required to approximate the integral to within $\displaystyle 10^-6$ by using the composite trapezoidal rule.