Mr. Integral
Math notes/Calculus II/Parametric and Polar Calculus

THE WHOLE UNIT · ONE REFERENCE

Parametric and Polar Calculus
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

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Key formulas, conditions and traps · Read down each column.

Parametric Curves and Motion

Core rule

dydx=y′x′,d2ydx2=(y′/x′)′x′,L=∫x′2+y′2 dt.\frac{dy}{dx}=\frac{y'}{x'},\quad \frac{d^2y}{dx^2}=\frac{(y'/x')'}{x'},\quad L=\int\sqrt{x'^2+y'^2}\,dt.

Watch for

Slope formulas require x′≠0x'\ne0. Distance counts repeated traversal, while displacement depends only on endpoints.

Polar Curves, Area and Length

Core rule

x=rcos⁡θ,y=rsin⁡θ,A=12∫r2dθ,L=∫r2+r′2dθ.x=r\cos\theta,\quad y=r\sin\theta,\quad A=\frac12\int r^2d\theta,\quad L=\int\sqrt{r^2+r'^2}d\theta.

Watch for

Choose an interval tracing the intended loop once. Negative radii reverse direction; squaring does not prevent double-counting.

Calculus II Checkpoint

Core rule

Select a method from structure: composition suggests substitution, products may suggest parts, factorials suggest ratios, and powers indexed by nn suggest roots.

Watch for

Check endpoints, hypotheses, units, geometric coverage and approximation error before accepting a computed answer.

01

Parametric Curves and Motion

2 reference blocks

Read lesson ↗

Core rule

dydx=y′x′,d2ydx2=(y′/x′)′x′,L=∫x′2+y′2 dt.\frac{dy}{dx}=\frac{y'}{x'},\quad \frac{d^2y}{dx^2}=\frac{(y'/x')'}{x'},\quad L=\int\sqrt{x'^2+y'^2}\,dt.

Watch for

Slope formulas require x′≠0x'\ne0. Distance counts repeated traversal, while displacement depends only on endpoints.

02

Polar Curves, Area and Length

2 reference blocks

Read lesson ↗

Core rule

x=rcos⁡θ,y=rsin⁡θ,A=12∫r2dθ,L=∫r2+r′2dθ.x=r\cos\theta,\quad y=r\sin\theta,\quad A=\frac12\int r^2d\theta,\quad L=\int\sqrt{r^2+r'^2}d\theta.

Watch for

Choose an interval tracing the intended loop once. Negative radii reverse direction; squaring does not prevent double-counting.

03

Calculus II Checkpoint

2 reference blocks

Read lesson ↗

Core rule

Select a method from structure: composition suggests substitution, products may suggest parts, factorials suggest ratios, and powers indexed by nn suggest roots.

Watch for

Check endpoints, hypotheses, units, geometric coverage and approximation error before accepting a computed answer.