Section 3 Arc Length

Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above parametric equations on the interval [t 1,t 2 ] is defined to be:

Example (1) Find the arc length L of the curve: x=rcost, y=rsint ; 0 ≤ t ≤ 2 π, where r is a constant ( what does that represent?) Solution:

Example (2) Find the arc length L of the curve: x=r(t-sint), y=r(1-cost) ; 0 ≤ t ≤ 2 π, where r is a constant Solution: dx/dt = ? dy/dt=?

Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] Assume that g is non-negative and the derivatives of f and g are continuos on that interval. The surface area resulting from the rotation the curve having the above parametric equations about the x-axis on the interval [t 1,t 2 ] is defined to be:

Example (1) Find the surface area of the sphere with radius r ! Solution: Consider the semicircle: x = r cost, y = r sint ; 0 ≤ t ≤ π, where r is a constant The rotation of this semicircle results in a sphere of radius r