Exam 10: Parametric Equations and Polar Coordinates

Find the exact area of the surface obtained by rotating the given curve about the x-axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

Find an equation for the conic that satisfies the given conditions. parabola, vertex (0, 0), focus (0, - $4$ )

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = 4 t \cos t , y = 4 t \sin t , t = - \pi$

Find the eccentricity of the conic. $r = \frac { 7 } { 8 - 7 \sin \theta }$

The curve $x = 2 - 4 \cos ^ { 2 } t , \quad y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)$ cross itself at some point $\left( x _ { 0 } , y _ { 0 } \right)$ . Find the equations of both tangent lines at that point.

Consider the polar equation $r = \frac { 9 } { 5 + 3 \sin \theta }$ . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus. Its perihelion distance (minimum distance from the planet to the sun) is approximately 1.3 $\times 10 ^ { 7 }$ km, and its aphelion distance (maximum distance from the planet to the sun) is approximately 6.9 $\times 10 ^ { 7 }$ km. Approximate the eccentricity of the planet's orbit. Round to three decimal places.

If a and b are fixed numbers, find parametric equations for the set of all points P determined as shown in the figure, using the angle ang as the parameter. Write the equations for $a = 12$ and $b = 4$ .

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = t \cos t , \quad y = t \sin t , \quad t = 5 \pi$

Find the vertices, foci, and asymptotes of the hyperbola. $y ^ { 2 } - 5 x ^ { 2 } = 25$

If a projectile is fired with an initial velocity of $\mathcal { v } _ { 0 }$ meters per second at an angle $\alpha$ above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t$ , $y = \left( v _ { 0 } \sin \alpha \right) t - \frac { 1 } { 2 } g t ^ { 2 }$ , where g is the acceleration of gravity $\left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$ . If a gun is fired with $\alpha = 55 ^ { \circ }$ and $\nu _ { 0 } = 440 \mathrm {~m} / \mathrm { s }$ when will the bullet hit the ground?

Find the area of the region that lies inside both curves. $r = 8 + 2 \sin \theta , r = 7$

Find the length of the polar curve. $r = 3 \cos \theta , 0 \leq \theta \leq \frac { 3 \pi } { 4 }$