What can you say about the arc length of the portion of the curve

$$r=1 / \theta$$

for

$$\pi / 4 \leq \theta \leq \pi / 2$$

that lies inside the circle r=1?

Use the formula below to calculate the arc length of the polar curve.

Formula:

$$L=\int_\alpha^\beta\sqrt{[f(\theta)]^2+[f'(\theta)]^2}d\theta=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$

The entire cardioid r = a(1 − cos theta)

Graph the equation using a graphing utility.

$$x=y+2 y ^ { 3 } - y ^ { 5 }$$

Determine whether the statement is true or false. Explain your answer. If d is a positive constant, then the conic section with polar equation

$$r = \frac { d } { 1 + \cos \theta }$$

is a parabola.

Show that the curve $x = t ^ { 3 } - 4 t , y = t ^ { 2 }$ intersects itself at the point $( 0,4 ) ,$ and find equations for the two tangent lines to the curve at the point of intersection.

Show that the graph of the given equation is a parabola. Find its vertex, focus, and directrix.

$$x ^ { 2 } + 2 \sqrt { 3 } x y + 3 y ^ { 2 } + 16 \sqrt { 3 } x - 16 y - 96 = 0$$

Determine whether the statement is true or false. Explain your answer.

If a conic $r=\frac{e d}{1+e \cos \theta}$ is a hyperbola then the polar function has two discontinuities in the interval $0 \leq \theta \leq 2 \pi$.

A parametric curve of the form x=a cot t+b cos t, y=a+b sin t (0<t<2pi) is called a conchoid of Nicomedes.

Find a polar equation r=f(theta) for the conchoid in part (a), and then find polar equations for the tangent lines to the conchoid at the pole.

A parametric curve of the form x=a cot t+b cos t, y=a+b sin t (0<t<2pi) is called a conchoid of Nicomedes.

For what values of t does the conchoid in part (a) have a horizontal tangent line? A vertical tangent line?

A parametric curve of the form x=a cot t+b cos t, y=a+b sin t (0<t<2pi) is called a conchoid of Nicomedes.

Find the horizontal asymptote of the conchoid given in part (a).

Graph the equation using a graphing utility.

$$x=sin y, - 2 \pi \leq y \leq 2 \pi$$

Use the formula below to calculate the arc length of the polar curve.

Formula:

$$L=\int_\alpha^\beta\sqrt{[f(\theta)]^2+[f'(\theta)]^2}d\theta=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$

The entire circle r = 2a cos theta

Graph the equation using a graphing utility.

$x=\sin y,-2 \pi \leq y \leq 2 \pi$

Find an equation for the ellipse that satisfies the given conditions. Foci (-1, 1) and (-1, 3); minor axis of length 4.