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)

Find an equation for the given polar graph.

Explain why y can be expressed as a differentiable periodic function of x. What is the period of this function?

How high must the ceiling be to ensure that the plane does not touch or crash into it? x=t-2 sin t, y=3-2 cos t

Find an equation for the given polar graph.

(a) Show that $r = \sin \theta + \cos \theta$ is a circle. (b) Find the area of the circle using a geometric formula and then by integration.

Assume that the plane flies in a room in which the floor is at y=0. How close to the floor does the airplane get? How far from the floor does the airplane get?

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.

Use a graphing utility to generate the trajectory of a paper airplane whose equations of motion for $t \geq 0$ are

$$x=2 t-\sin t, \quad y=6-3 \cos t$$

In Exercise, find the rectangular coordinates of the points whose polar coordinates are given.

$$\begin{array} { l l } { \text { (a) } ( - 8 , \pi / 4 ) } & { \text { (b) } ( 7 , - \pi / 4 ) } & { \text { (c) } ( 8,9 \pi / 4 ) } \\ { \text { (d) } ( 5,0 ) } & { \text { (e) } ( - 2 , - 3 \pi / 2 ) } & { \text { (f) } ( 0 , \pi ) } \end{array}$$

Assuming that the plane flies in a room in which the floor is at y=0, explain why the plane will not crash into the floor. [For simplicity, ignore the physical size of the plane by treating it as a particle.] x=t-2 sin t, y=3-2 cos t

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?

Use a graphing utility to generate the trajectory of a paper airplane whose equations of motion for

$$t \geq 0$$

are x=t-2 sin t, y=3-2 cos t.

Find an equation for the ellipse that satisfies the given conditions.

. Foci $(0, \pm 2) ; \quad b=2 \sqrt{2}$