zuai-logo

Arc length formula for parametric curves?

S=∫ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt

All Flashcards

Arc length formula for parametric curves?
$S=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt$
Arc length formula for Cartesian curves?
$S=\int_a^b \sqrt{1+[f'(x)]^2} dx$
Define arc length.
The distance between two points along a curve.
What is a parametric curve?
A curve where the x and y coordinates are defined by functions of a parameter, usually denoted as t.
Define a smooth planar curve.
A curve in a plane that has a continuous first derivative.
Steps to find arc length of parametric curve?
1. Find dx/dt and dy/dt. 2. Substitute into the arc length formula. 3. Evaluate the integral.
How to evaluate $\int_0^\pi \sqrt{1} dt$?
1. Integrate to get [t]. 2. Evaluate at the bounds: $\pi - 0 = \pi$.