GMP Unit 3

Parametric and Polar Equations and Calculas

$\frac{dy}{dx}$ of parametric equations

$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

length of an arc of parametric equations

\int_{a}^{b}\sqrt{\left(\frac{dx}{\differentialD t}\right)^2+\left(\frac{dy}{\differentialD t}\right)^2}\!\,dt

Area enclosed by two curves in parametric equations =

$$\int_{a}^{b}g(t)f^{\prime}(t)dt\,$$

polar equations (x =)

$$X=r\cos\left(\theta\right)$$

Polar equations (y=)

$$Y=r\sin\left(\theta\right)$$

\frac{dy}{\differentialD x}= (of polar equations)

$$\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{r\cos\theta+\sin\theta\left(\frac{dr}{d\theta}\right)}{-r\sin\theta+\cos\theta\left(\frac{dr}{d\theta}\right)}$$

area of polar curves:

$$\frac12\int_{a}^{b}r^2d\theta$$

Area between two graphs of polar coordinates

$\frac12\int_{a}^{b}\left(R^2-r^2\right)d\theta$ ra

arc length of polar curves

$$\int_{a}^{b}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$