Calculus 2: Polar coordinates

Find the area enclosed by $x^2 + y^2 = r^2$.

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

Plot and identify the positions of three points given in polar coordinates.

Convert a point from polar to rectangular coordinates using trigonometry and Pythagorean theorem.

Convert a rectangular equation, such as $x + y = 3$, to a polar equation.

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

Graph polar coordinates with given radius and angle.

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.