Section 9.11
Comparison of Methods
One geometry, three languages. Choosing between Cartesian, Parametric, or Polar isn't just about preference—it's about making the calculus possible.
1
The Rosetta Stone
System Translation Guide
The physical reality (length, area) is invariant—it doesn't care what coordinates you use. But the algebra cares a lot.
| System | Variables | Arc Length () Element | Area Element |
|---|---|---|---|
| Cartesian | (Rectangles) | ||
| Parametric | (Rectangles) | ||
| Polar | (Wedges) |
2
Worked Example
Circle Circumference Three Ways
Show circumference is .
Cartesian
. Nasty derivative. Hard triangle sub integral.Parametric
. .. Easy!
Polar
. .. Easiest!
3
Level Up Examples
A. The Parametric Win
The Cycloid: .
Why Cartesian Fails
Solving for in terms of involves infinite inverse trig functions. Integrating in is nearly impossible.
Parametric Power
.
Solvable with simple trig identities!
Smooth parameterization makes the math behave.
B. The Polar Win
The Rose: .
Why Cartesian Fails
Try writing in standard form. Integrating that area is a nightmare.
Polar Power
Bounds: to .
.
One simple integral.
Polar coordinates respect the rotational symmetry.
5
Practice Quiz
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