# Active Calculus

## Section9.5Calculus in Polar Coordinates

### ExercisesExercises

#### 1.

(a) The Cartesian coordinates of a point are $$(1,1).$$
(i) Find polar coordinates $$(r,\theta)$$ of the point, where $$r>0$$ and $$0 \le \theta \lt 2\pi.$$
$$r =$$
$$\theta =$$
(ii) Find polar coordinates $$(r,\theta)$$ of the point, where $$r\lt 0$$ and $$0 \le \theta \lt 2\pi.$$
$$r =$$
$$\theta =$$
(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 $$0 \le \theta \lt 2\pi.$$
$$r =$$
$$\theta =$$
(ii) Find polar coordinates $$(r,\theta)$$ of the point, where $$r\lt 0$$ and $$0 \le \theta \lt 2\pi.$$
$$r =$$
$$\theta =$$

#### 2.

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

#### 3.

Find the equation in polar coordinates of the line through the origin with slope $$\frac1{5}\text{.}$$
$$\theta=$$

#### 4.

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

#### 5.

Find the equation (in terms of $$x$$ and $$y$$) of the tangent line to the curve $$r=4\sin5\theta$$ at $$\theta=\pi/4\text{.}$$
$$y =$$

#### 6.

Find the area of the region bounded by the polar curve $$r = 9 e^ \theta$$ , on the interval $$\frac{4}{9} \pi \leq \theta \leq 2 \pi\text{.}$$

#### 7.

Find the total area enclosed by the cardioid $$r=8-\cos\theta$$ shown in the following figure:

#### 8.

Find the area of one leaf of the "four-petaled rose" $$r =9 \sin 2\theta$$ shown in the following figure:
With $$r_0=9$$

#### 9.

Find the area lying outside $$r=4\sin\theta$$ and inside $$r=2 + 2 \sin\theta\text{.}$$
Area =

#### 10.

Find the exact length of the polar curve
\begin{equation*} r=3\sin(\theta),\;\;0 \le \theta \le \pi/3. \end{equation*}
Length =

#### 11.

Find the length of the spiraling polar curve
$$r = 4 e^{6 \theta}$$
From 0 to $$2 \pi$$ .
The length is