Calculus 2: Polar coordinates

Convert Cartesian coordinates to polar coordinates and sketch the polar curve for $r = \cos(2\theta)$.

Graph the polar equation $r = 2 \cos(\theta)$, and verify by converting to rectangular form.

Find the area of a circle using polar coordinates.

Graph the equation where $r = a \cos(2\theta)$.

Find the equation of the tangent line for the polar equation $r = 2 + 3\cos(\theta)$ when $\theta = \frac{\pi}{2}$.

Write the equation of the tangent line for the polar equation $r = 3 - 2\sin(\theta)$ when $\theta = \pi$.

Evaluate the double integral $\displaystyle \int_{0}^{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-r^2} r \, dr \, d\theta$ by converting it to polar coordinates in the region bounded by $x = \sqrt{4 - y^2}$ and the y-axis.