Parametric equation ?

Question

Consider the parametric equation

x = 9(cosx+xsinx)

y = 9(sinx-xcosx)

What is the length of the curve for x= 0 to x=3pi/10

Answer 1

1. This is not a parametric equation. You define y as a function of x but then have an equation of x in terms of itself. This equation has an infinite number of solutions, but they are discrete, not continuous.

2. x = 9(cos(t) + t sin(t) would be a reasonable parametric equation, but then would the equation for y be:

y = 9(sin x - x cos x) or

y = 9(sin t - t cos t)

(I'd guess the latter, but who can tell? And even if I could, there would still be the question of whether the interval is x = 0 to x=3pi/10 or t = 0 to t=3pi/10)

3. The standard way to compute the arc length is by integration:

http://en.wikipedia.org/wiki/Arc_length

In this case, the integral should be simplified by some trigonometric facts:

sin^2(A) + cos^2(A) = 1

d(cos(t))/dt = -sin(t), d(sin(t))/dt = cos(t)