7.4 Area and arc length in polar coordinates  (Page 3/3)

Region enclosed by $r=4$

Region enclosed by $r=3\text{sin}\theta$

Region in the first quadrant within the cardioid $r=1+\text{sin}\theta$

Region enclosed by one petal of $r=8\text{sin}(2\theta )$

Region enclosed by one petal of $r=\text{cos}(3\theta )$

Region below the polar axis and enclosed by $r=1-\text{sin}\theta$

Region in the first quadrant enclosed by $r=2-\text{cos}\theta$

Region enclosed by the inner loop of $r=2-3\text{sin}\theta$

Region enclosed by the inner loop of $r=3-4\text{cos}\theta$

Region enclosed by $r=1-2\text{cos}\theta$ and outside the inner loop

Region common to $r=3\text{sin}\theta \text{and}r=2-\text{sin}\theta$

Region common to $r=2\text{and}r=4\text{cos}\theta$

Region common to $r=3\text{cos}\theta \text{and}r=3\text{sin}\theta$

Enclosed by $r=6\text{sin}\theta$

Above the polar axis enclosed by $r=2+\text{sin}\theta$

Below the polar axis and enclosed by $r=2-\text{cos}\theta$

Enclosed by one petal of $r=4\text{cos}\left(3\theta \right)$

Enclosed by one petal of $r=3\text{cos}\left(2\theta \right)$

Enclosed by $r=1+\text{sin}\theta$

Enclosed by the inner loop of $r=3+6\text{cos}\theta$

Enclosed by $r=2+4\text{cos}\theta$ and outside the inner loop

Common interior of $r=4\text{sin}\left(2\theta \right)\text{and}r=2$

Common interior of $r=3-2\text{sin}\theta \text{and}r=\mathrm{-3}+2\text{sin}\theta$

Common interior of $r=6\text{sin}\theta \text{and}r=3$

Inside $r=1+\text{cos}\theta$ and outside $r=\text{cos}\theta$

Common interior of $r=2+2\text{cos}\theta \text{and}r=2\text{sin}\theta$

$r=4\text{cos}\theta \text{on the interval}0\le \theta \le \frac{\pi }{2}$

$r=1+\text{sin}\theta$ on the interval $0\le \theta \le 2\pi$

$r=2\text{sec}\theta \text{on the interval}0\le \theta \le \frac{\pi }{3}$

$r=e^{\theta }\text{on the interval}0\le \theta \le 1$

$r=6\text{on the interval}0\le \theta \le \frac{\pi }{2}$

$r=e^{3\theta }\text{on the interval}0\le \theta \le 2$

$r=6\text{cos}\theta \text{on the interval}0\le \theta \le \frac{\pi }{2}$

$r=8+8\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

$r=1-\text{sin}\theta \text{on the interval}0\le \theta \le 2\pi$

[T] $r=3\theta \text{on the interval}0\le \theta \le \frac{\pi }{2}$

[T] $r=\frac{2}{\theta }\text{on the interval}\pi \le \theta \le 2\pi$

[T] $r={\text{sin}}^2\left(\frac{\theta }{2}\right)\text{on the interval}0\le \theta \le \pi$

[T] $r=2{\theta }^2\text{on the interval}0\le \theta \le \pi$

[T] $r=\text{sin}\left(3\text{cos}\theta \right)\text{on the interval}0\le \theta \le \pi$

$r=3\text{sin}\theta \text{on the interval}0\le \theta \le \pi$

$r=\text{sin}\theta +\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

$r=6\text{sin}\theta +8\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

$r=3\text{sin}\theta \text{on the interval}0\le \theta \le \pi$

$r=\text{sin}\theta +\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

$r=6\text{sin}\theta +8\text{cos}\theta \text{on the interval}0\le \theta \le \pi$

Verify that if $y=r\text{sin}\theta =f(\theta )\text{sin}\theta$ then $\frac{dy}{d\theta }=f\prime (\theta )\text{sin}\theta +f(\theta )\text{cos}\theta .$

Use the definition of the derivative $\frac{dy}{dx}=\frac{dy\text{/}d\theta }{dx\text{/}d\theta }$ and the product rule to derive the derivative of a polar equation.

$r=1-\text{sin}\theta ;$ $\left(\frac{1}{2},\frac{\pi }{6}\right)$

$r=4\text{cos}\theta ;$ $\left(2,\frac{\pi }{3}\right)$

$r=8\text{sin}\theta ;$ $\left(4,\frac{5\pi }{6}\right)$

$r=4+\text{sin}\theta ;$ $\left(3,\frac{3\pi }{2}\right)$

$r=6+3\text{cos}\theta ;$ $\left(3,\pi \right)$

$r=4\text{cos}\left(2\theta \right);$ tips of the leaves

$r=2\text{sin}\left(3\theta \right);$ tips of the leaves

$r=2\theta ;$ $\left(\frac{\pi }{2},\frac{\pi }{4}\right)$

Find the points on the interval $\text{−}\pi \le \theta \le \pi$ at which the cardioid $r=1-\text{cos}\theta$ has a vertical or horizontal tangent line.

For the cardioid $r=1+\text{sin}\theta ,$ find the slope of the tangent line when $\theta =\frac{\pi }{3}.$

$r=3\text{cos}\theta ,\theta =\frac{\pi }{3}$

$r=\theta ,$ $\theta =\frac{\pi }{2}$

$r=\text{ln}\theta ,$ $\theta =e$

[T] Use technology: $r=2+4\text{cos}\theta$ at $\theta =\frac{\pi }{6}$

$r=4\text{cos}\theta$

$r^2=4\text{cos}\left(2\theta \right)$

$r=2\text{sin}(2\theta )$

The cardioid $r=1+\text{sin}\theta$

Show that the curve $r=\text{sin}\theta \text{tan}\theta$ (called a cissoid of Diocles ) has the line $x=1$ as a vertical asymptote.