Exam 10: Parametric Equations and Polar Coordinates

Set up, but do not evaluate, an integral that represents the length of the parametric curve. $x = t - t ^ { 10 } , y = \frac { 10 } { 9 } t ^ { 9 / 8 } , 8 \leq \mathrm { t } \leq 18$

Find the area of the region that is bounded by the given curve and lies in the specified sector. $r = 9 \sqrt { \sin 2 \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

A cow is tied to a silo with radius $8$ by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow. Round the answer to the nearest hundredth.

The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity $0.995$ and one focus at the Sun. The length of its major axis is $369.9$ AU. [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun. (The perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ .) Find the answer in AU and round to the nearest hundredth.

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. $x = 3 \sin t , \quad y = 3 \sin 2 t , \quad 0 \leq t \leq \pi / 2$

Consider the polar equation $r = - \frac { 7 } { 2 \sin \theta - 1 }$ . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Find an equation of the conic satisfying the given conditions. Hyperbola, foci (5, 6) and (5, -2), asymptotes x = 2y + $1$ and x = - 2y + $9$

Find an equation of the tangent line to the curve at the point corresponding to the value of the parameter. $x = e ^ { \sqrt { t } }$ , $y = t - \ln t ^ { 6 }$ ; $t = 1$

Use a graph to estimate the values of $\theta$ for which the curves $r = 9 + 3 \sin 5 \theta$ and $r = 18 \sin \theta$ intersect. Round your answer to two decimal places.

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . $x = 4 ( t + \sin t ) , y = 4 ( t - \cos t )$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = \cos \theta + \sin 2 \theta + 8 , \quad y = \sin \theta + \cos 2 \theta + 8 , \quad \theta = \pi$

Graph of the following curve is given. Find its length. $r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$

Find the polar equation for the curve represented by the given Cartesian equation. $x + y = 2$

Eliminate the parameter to find a Cartesian equation of the curve. $x ( t ) = 2 \cos ^ { 2 } t , \quad y ( t ) = 7 \sin ^ { 2 } t$

Find the surface area generated by rotating the lemniscate $r ^ { 2 } = 10 \cos 2 \theta$ about the line $\theta = \pi$ .

Write a polar equation in r and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 2 } { 7 }$ and directrix $x = - 11$ .

True or False? If the parametric curve x = f ( $f$ ), y = g ( $t$ ) satisfies g '( $4$ ) = 0, then it has a horizontal tangent when $t$ = $4$ .

Find a Cartesian equation for the curve described by the given polar equation. $r = 11 \sin \theta$

Write a polar equation in r and $\theta$ $\theta$ of an ellipse with the focus at the origin, with the eccentricity $0.2$ and vertex at $\left( 1 , \frac { \pi } { 2 } \right)$ .

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , r = 1 + \cos \theta$