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 \(0 \le \theta \lt 2\pi.\)

\(r =\)

\(\theta =\)

(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 \(0 \le \theta \lt 2\pi.\)

\(r =\)

\(\theta =\)

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 \(\frac1{5}\text{.}\)

\(\theta=\)

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 \(r=4\sin5\theta\) at \(\theta=\pi/4\text{.}\)

\(y =\)

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

Answer:

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

Answer :

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

With \(r_0=9\)

Answer :

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

Area =

Length =

Find the length of the spiraling polar curve

From 0 to \(2 \pi\) .

The length is