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Wize AP Calculus (BC) Textbook > Parametric & Polar Curves

Calculus w/ Polar Coordinates

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Tangents of Polar Curves (Derivatives)

First Derivative

(a.k.a slope of the tangent to the curve)
dydx=drdθsin⁡θ+rcos⁡θdrdθcos⁡θ−rsin⁡θ   or   y′(θ)x′(θ)\displaystyle \frac{dy}{dx}=\frac{\displaystyle \frac{dr}{d\theta}\sin\theta+r\cos\theta}{\displaystyle \frac{dr}{d\theta}\cos\theta-r\sin\theta}\ \ \ \text{or}\ \ \ \frac{y'\left(\theta\right)}{x'\left(\theta\right)}
*Recall that y=rsin⁡θy=r\sin\theta, x=rcos⁡θx=r\cos\theta

The polar curve has
  • A Horizontal Tangent when
  • drdθsin⁡θ+rcos⁡θ   or   y′(θ)=0\displaystyle \frac{dr}{d\theta}\sin\theta+r\cos\theta\ \ \ \text{or}\ \ \ y'\left(\theta\right)=0 and
  • drdθcos⁡θ−rsin⁡θ   or   x′(θ)≠0\displaystyle \frac{dr}{d\theta}\cos\theta-r\sin\theta\ \ \ \text{or}\ \ \ x'\left(\theta\right)\ne0
  • A Vertical Tangent when
  • drdθcos⁡θ−rsin⁡θ   or   x′(θ)=0\displaystyle \frac{dr}{d\theta}\cos\theta-r\sin\theta\ \ \ \text{or}\ \ \ x'\left(\theta\right)=0 and
  • drdθsin⁡θ+rcos⁡θ   or   y′(θ)≠0\displaystyle \frac{dr}{d\theta}\sin\theta+r\cos\theta\ \ \ \text{or}\ \ \ y'\left(\theta\right)\ne0
*If both y′(θ)=0y'\left(\theta\right)=0 and x′(θ)=0x'\left(\theta\right)=0, use L'Hosputal's rule

Practice Question

Find the slope of the tangent line to the curve r=sec⁡2θcsc⁡θr = \sec^2 \theta \csc\theta at θ=π/6.\theta = \pi/6.

Practice: Tangents

Consider the polar curve r=eθr=e^{\theta} where θ∈[0,2π]\theta\in\left[0,2\pi\right].
a) Find the slope of the tangent line to the curve at θ=π\theta=\pi.
b) Find all points at which the tangent line to the curve is horizontal.
c) Find all points at which the tangent line to the curve is vertical.
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Applications of Polar Curves

Consider the polar curve r=r(θ)r=r\left(\theta\right) where a≤θ≤ba\le\theta\le b. Suppose that f′(θ)f'\left(\theta\right) is continuous on [a,b]\left[a,b\right] and the curve is traversed exactly once on [a,b]\left[a,b\right].

Arc Length L=∫ab(r(θ))2+r′(θ)2  ⁣dθL={\displaystyle\int}^b_a \sqrt{\left(r(\theta)\right)^2+r'(\theta )^2}\de{\theta}


Areas Under the Curve A=∫ab12[r(θ)]2  ⁣dθA={\displaystyle\int}^b_a\frac{1}{2}[r(\theta)]^2\de{\theta}


Area Bounded Between Two Polar Curves

A=12∫ab[(router)2−(rinner)2]  ⁣dθA=\frac{1}{2}{\displaystyle\int}^b_a[(r_{outer})^2-(r_{inner})^2]\de{\theta}


Wize Tip
Always graph the curves before using the formula!

Practice: Arc Length

Compute the total length of the curve r=2acos⁡θr = 2a \cos \theta for a>0a>0 and θ∈[0,π]\theta\in\left[0,\pi\right]

Practice: Area Under Polar Curve

Find the area inside r=1+sin⁡θr=1+\sin\theta.

Practice Question

Write the integral representing the area of the inner loop of r=2+4sin⁡θ.r = 2 + 4 \sin \theta.

Practice: Area Between Polar Curves

Find the area of the region lying inside the limacon curve of equation r=1+2cosθr = 1 + 2 cos \theta, but outside the circle rr = 2.