How does arc length relate to the Pythagorean Theorem?
Arc length is approximated by summing the hypotenuses of small right triangles with sides dx and dy.
Why do we need a different arc length formula for parametric curves?
Because both x and y coordinates are changing with respect to a parameter t, so we must account for both rates of change.
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.
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?
$S=\int_a^b \sqrt{1+[f'(x)]^2} dx$
