Calculus II - Polar equations

4 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Polar form

   In polar form, points of a graph (and therefore of a function) are defined in terms of distance, rrr , and angle from the pole, θ\thetaθ . This leads to functions commonly being represented in the form r=f(θ)r=f(\theta)r=f(θ)

2. Find parametric form of polar curve

   Simply substitute the polar equation into the parametric equations below: x=rcos⁡θx=r\cos\thetax=rcosθ , y=rsin⁡θy=r\sin\thetay=rsinθ

3. Arc length of a polar equation

   If a curve CCC is represented by the polar equation r=f(θ)r=f(\theta)r=f(θ) , α≤0≤β\alpha \leq 0 \leq \betaα≤0≤β , and if f′(θ)=drdθf'(\theta)=\frac{dr}{d\theta}f′(θ)=dθdr​ is continuous on an interval containing α\alphaα and β\betaβ , then the arc length sss of CCC from θ=α\theta=\alphaθ=α to θ=β\theta=\betaθ=β is s=∫abr2+(drdθ)2dθs=\int_a^b\sqrt{r^2+(\frac{dr}{d\theta})^2}d\thetas=∫ab​r2+(dθdr​)2 ​dθ

4. Area of a polar equation

   If r=f(θ)r=f(\theta)r=f(θ) is nonnegative and continuous on the closed interval [α,β][\alpha, \beta][α,β] , where α<β\alpha < \betaα<β and β−α≤2π\beta-\alpha \leq 2\piβ−α≤2π , then the area AAA of the region bounded by the graph of r=f(θ)r=f(\theta)r=f(θ) and the rays θ=α\theta=\alphaθ=α and θ=β\theta=\betaθ=β is given by A=∫αβ12r2dθA=\int_\alpha^\beta \frac{1}{2}r^2d\thetaA=∫αβ​21​r2dθ