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Formula sheet

Parametric and polar — Formula Sheet

3 formulas from Calculus II, each with what it means, when to use it and the trap it sets.

Slope of a parametric curve

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

What it says: Both coordinates depend on t, and the dt cancels between them.
When to reach for it: x and y are each given as a function of a parameter.
Watch out: Divide the derivatives; do not differentiate the quotient y/x. Vertical tangents happen where dx/dt = 0.

x = t^{2}, y = t^{3} \Rightarrow \frac{d y}{d x} = \frac{3 t ^{2}}{2 t} = \frac{3 t}{2}

The answer stays in terms of t, which is fine.

Parametric arc length

L = \int_{a}^{b} (\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2} d t

What it says: Pythagoras on an infinitesimal step, added along the curve.
When to reach for it: The length of a parametric path, or of y = f(x) with x as its own parameter.
Watch out: The square root covers the sum of both squares, and the limits are values of t, not of x.

L = \int_{0}^{2 π} sin^{2} t + cos^{2} t d t = 2 π

The unit circle — the identity collapses the root to 1.

Area in polar coordinates

A = \frac{1}{2} \int_{α}^{β} r^{2} d θ

What it says: Sweep out thin triangles from the origin and add their areas.
When to reach for it: A region described by r = f(θ).
Watch out: The ½ and the square are both essential, and the limits are the angles where the curve actually traces the region once.

A = \frac{1}{2} \int_{0}^{2 π} a^{2} d θ = π a^{2}

The circle r = a, recovering the familiar area.

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