Active Calculus

(a) The Cartesian coordinates of a point are $(1,1).$

(i) Find polar coordinates $(r,\theta )$ of the point, where $r>0$ and $0\le \theta <2\pi .$

$r=$

$\theta =$

(ii) Find polar coordinates $(r,\theta )$ of the point, where $r<0$ and $0\le \theta <2\pi .$

$r=$

$\theta =$

(b) The Cartesian coordinates of a point are $(2\sqrt{3},-2).$

(i) Find polar coordinates $(r,\theta )$ of the point, where $r>0$ and $0\le \theta <2\pi .$

$r=$

$\theta =$

(ii) Find polar coordinates $(r,\theta )$ of the point, where $r<0$ and $0\le \theta <2\pi .$

$r=$

$\theta =$

For each set of Polar coordinates, match the equivalent Cartesian coordinates.

Find the equation in polar coordinates of the line through the origin with slope $\frac{1}{4}\text{.}$

$\theta =$

Find the slope of the tangent line to the polar curve $r=1/\theta$ at the point specified by $\theta =\pi .$

Slope =

Find the equation (in terms of $x$ and $y$) of the tangent line to the curve $r=3\sin 4\theta$ at $\theta =\pi /6\text{.}$

$y=$

Find the area of the region bounded by the polar curve $r=7e^{\theta }$ , on the interval $\frac{4}{7}\pi \le \theta \le 2\pi \text{.}$

Answer:

Find the total area enclosed by the cardioid $r=7-\cos \theta$ shown in the following figure:

Answer :

Find the area of one leaf of the "four-petaled rose" $r=8\sin 2\theta$ shown in the following figure:

With $r_0=8$

Answer :

Find the area lying outside $r=2\cos \theta$ and inside $r=1+\cos \theta \text{.}$

Area =

Length =

Find the length of the spiraling polar curve

$r=3e^{5\theta }$

From 0 to $2\pi$ .

The length is