Calculus II delves into advanced integration techniques, series, and applications of calculus to real-world problems.
We usually use \( y = f(x) \) to describe curves, but parametric and polar equations give you new perspectives!
Use a third variable (usually \( t \)) to describe both \( x \) and \( y \):
This is great for describing motion, like a planet orbiting the sun.
Instead of \( (x, y) \), describe points as \( (r, \theta) \)—distance from origin and angle. This is super useful for circles and spirals.
These approaches make it easier to analyze curves that are hard or impossible to describe using standard \( x \) and \( y \) equations.
Describe the motion of a particle moving in a circle using \( x = \cos t, y = \sin t \).
Sketch the polar curve \( r = 2 + \cos \theta \).
Parametric and polar equations give you fresh ways to describe and analyze curves.