Question 1

Write the corresponding rectangular equation for the curve represented by the parametric equations $x = 9 \cos \theta , y = 9 \sin \theta$ 
by eliminating the parameter.

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = 18$ 
B)  $x + y = 9$ 
C)  $x + y = 18$ 
D)  $x ^ { 2 } + y ^ { 2 } = 9$ 
E)  $x ^ { 2 } + y ^ { 2 } = 81$ 
A)  $x ^ { 2 } + y ^ { 2 } = 18$ 
B)  $x + y = 9$ 
C)  $x + y = 18$ 
D)  $x ^ { 2 } + y ^ { 2 } = 9$ 
E)  $x ^ { 2 } + y ^ { 2 } = 81$

Question 2

Find  a polar equation for the parabola with its focus at the pole and vertex $$\left( 3 , - \frac { 1 } { 2 } \pi \right)$$

Accepted Answer

A) $r = \frac { 6 } { 1 - \cos \theta }$ 
B)  $r = \frac { 6 } { 1 - \sin \theta }$ 
C)  $r = - \frac { 6 } { 1 + \cos \theta }$ 
D)  $r = \frac { 6 } { 1 + \cos \theta }$ 
E)  $r = \frac { 6 } { 1 + \sin \theta }$ 
A) $r = \frac { 6 } { 1 - \cos \theta }$ 
B)  $r = \frac { 6 } { 1 - \sin \theta }$ 
C)  $r = - \frac { 6 } { 1 + \cos \theta }$ 
D)  $r = \frac { 6 } { 1 + \cos \theta }$ 
E)  $r = \frac { 6 } { 1 + \sin \theta }$

Question 3

Find an equation of the ellipse with vertices $( 0,8 ) , ( 14,8 ) \text { and eccentricity } e = \frac { 2 } { 7 }$

Accepted Answer

A) 
$\frac { ( x - 7 ) ^ { 2 } } { 45 } + \frac { ( y - 8 ) ^ { 2 } } { 49 } = 1$ 
B) 
$\frac { ( x - 8 ) ^ { 2 } } { 49 } + \frac { ( y - 7 ) ^ { 2 } } { 45 } = 1$ 
C) 
$\frac { ( x - 7 ) ^ { 2 } } { 45 } - \frac { ( y - 8 ) ^ { 2 } } { 49 } = 1$ 
D) 
$\frac { ( x - 7 ) ^ { 2 } } { 49 } + \frac { ( y - 8 ) ^ { 2 } } { 45 } = 1$ 
E) 
$\frac { ( x - 7 ) ^ { 2 } } { 49 } - \frac { ( y - 8 ) ^ { 2 } } { 45 } = 1$ 
A) 
$\frac { ( x - 7 ) ^ { 2 } } { 45 } + \frac { ( y - 8 ) ^ { 2 } } { 49 } = 1$ 
B) 
$\frac { ( x - 8 ) ^ { 2 } } { 49 } + \frac { ( y - 7 ) ^ { 2 } } { 45 } = 1$ 
C) 
$\frac { ( x - 7 ) ^ { 2 } } { 45 } - \frac { ( y - 8 ) ^ { 2 } } { 49 } = 1$ 
D) 
$\frac { ( x - 7 ) ^ { 2 } } { 49 } + \frac { ( y - 8 ) ^ { 2 } } { 45 } = 1$ 
E) 
$\frac { ( x - 7 ) ^ { 2 } } { 49 } - \frac { ( y - 8 ) ^ { 2 } } { 45 } = 1$

Question 4

Find the area of the common interior of the polar equations $r = 4 ( 1 + \sin \theta )$ and $r = 4 ( 1 - \sin \theta )$ by sketching the graph of the equations using the graphing utility.

Accepted Answer

A)  $16 ( 3 \pi + 8 )$ 
B)  $8 ( 3 \pi - 8 )$ 
C)  $8 ( 3 \pi - 4 )$ 
D)  $8 ( \pi - 4 )$ 
E)  $16 ( 3 \pi - 8 )$ 
A)  $16 ( 3 \pi + 8 )$ 
B)  $8 ( 3 \pi - 8 )$ 
C)  $8 ( 3 \pi - 4 )$ 
D)  $8 ( \pi - 4 )$ 
E)  $16 ( 3 \pi - 8 )$

Question 5

Find  the area of the surface generated by revolving the curve about the given axis. $x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2$
(i) $x$-axis;(ii) $y$-axis

Accepted Answer

A) (i) $\frac { 27 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
B) (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 54 } { 5 } \pi$ 
C)  (i) $\frac { 27 } { 10 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
D)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
E)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 108 } { 5 } \pi$ 
A) (i) $\frac { 27 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
B) (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 54 } { 5 } \pi$ 
C)  (i) $\frac { 27 } { 10 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
D)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
E)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 108 } { 5 } \pi$

Question 6

Find the eccentricity of the polar equation $$r = - \frac { 40 } { 20 + 41 \sin \theta }$$

Accepted Answer

A)  $\frac { 41 } { 40 }$ 
B)  $\frac { 20 } { 41 }$ 
C)  $\frac { 1 } { 41 }$ 
D)  $\frac { 1 } { 20 }$ 
E)  $\frac { 41 } { 20 }$ 
A)  $\frac { 41 } { 40 }$ 
B)  $\frac { 20 } { 41 }$ 
C)  $\frac { 1 } { 41 }$ 
D)  $\frac { 1 } { 20 }$ 
E)  $\frac { 41 } { 20 }$

Question 7

Find the center, foci, vertices, and eccentricity of the ellipse. $$\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 9 } = 1$$

Accepted Answer

A)  center $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 ) ;$ foci $( 3 , - 6 ) , ( 7,2 )$; eccentricity: $e = \frac { 5 } { 4 }$ 
B)  center: $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 )$; foci: $( - 1 , - 2 ) , ( 7 , - 2 )$; eccentricity: $e = \frac { 2 } { 5 }$ 
C)  center $( 3 , - 2 )$; vertices: $( 3 , - 7 ) , ( 3,3 )$; foci: $( 3 , - 6 ) , ( 37,2 )$; eccentricity:e $= \frac { 4 } { 5 }$ 
D)  center: $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 )$; foci: $( - 1 , - 2 ) , ( 7 , - 2 )$; eccentricity: $e = \frac { 5 } { 2 }$ 
E)  center: $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 )$; foci: $( - 1 , - 2 ) , ( 7 , - 2 )$; eccentricity: $e = \frac { 4 } { 5 }$ 
A)  center $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 ) ;$ foci $( 3 , - 6 ) , ( 7,2 )$; eccentricity: $e = \frac { 5 } { 4 }$ 
B)  center: $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 )$; foci: $( - 1 , - 2 ) , ( 7 , - 2 )$; eccentricity: $e = \frac { 2 } { 5 }$ 
C)  center $( 3 , - 2 )$; vertices: $( 3 , - 7 ) , ( 3,3 )$; foci: $( 3 , - 6 ) , ( 37,2 )$; eccentricity:e $= \frac { 4 } { 5 }$ 
D)  center: $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 )$; foci: $( - 1 , - 2 ) , ( 7 , - 2 )$; eccentricity: $e = \frac { 5 } { 2 }$ 
E)  center: $( 3 , - 2 )$; vertices: $( - 2 , - 2 ) , ( 8 , - 2 )$; foci: $( - 1 , - 2 ) , ( 7 , - 2 )$; eccentricity: $e = \frac { 4 } { 5 }$

Question 8

Find the eccentricity of the polar equation $$r = \frac { 78 } { 26 + \cos \theta }$$

Accepted Answer

A)  78 
B)  $\frac { 1 } { 26 }$ 
C)  26 
D)  13 
E)  $\frac { 1 } { 78 }$ 
A)  78 
B)  $\frac { 1 } { 26 }$ 
C)  26 
D)  13 
E)  $\frac { 1 } { 78 }$

Question 9

Find a polar equation for the parabola with its focus at the pole, eccentricity e = 1, and directrix $x = - 8$.

Accepted Answer

A) $r = - \frac { 8 } { 1 - \cos \theta }$ 
B)  $r = \frac { 8 } { 1 - \cos \theta }$ 
C)  $r = \frac { 8 } { 1 + \cos \theta }$ 
D) $r = - \frac { 8 } { 1 - \sin \theta }$ 
E)  $r = \frac { 8 } { 1 + \sin \theta }$ 
A) $r = - \frac { 8 } { 1 - \cos \theta }$ 
B)  $r = \frac { 8 } { 1 - \cos \theta }$ 
C)  $r = \frac { 8 } { 1 + \cos \theta }$ 
D) $r = - \frac { 8 } { 1 - \sin \theta }$ 
E)  $r = \frac { 8 } { 1 + \sin \theta }$

Question 10

Find the distance from the pole to the directrix for the conic $$r ( 16 + \sin \theta ) = 32$$

Accepted Answer

A)  $\frac { 1 } { 16 }$ 
B)  32 
C)  16 
D)  $\frac { 1 } { 32 }$ 
E)  8 
A)  $\frac { 1 } { 16 }$ 
B)  32 
C)  16 
D)  $\frac { 1 } { 32 }$ 
E)  8

Question 11

The path of a projectile is modeled by the parametric equations $x = \left( 85 \cos 60 ^ { \circ } \right) t \text { and }$ $y = \left( 85 \sin 60 ^ { \circ } \right) t - 16 t ^ { 2 }$ where $x$ and $y$ are measured in feet. Use a graphing utility to approximate the range of the projectile. Round your answer to two decimal places.

Accepted Answer

A)  $312.85 \mathrm { ft }$ 
B)  $521.42 \mathrm { ft }$ 
C)  $131.09 \mathrm { ft }$ 
D)  $391.06 \mathrm { ft }$ 
E)  $195.53 \mathrm { ft }$ 
A)  $312.85 \mathrm { ft }$ 
B)  $521.42 \mathrm { ft }$ 
C)  $131.09 \mathrm { ft }$ 
D)  $391.06 \mathrm { ft }$ 
E)  $195.53 \mathrm { ft }$

Question 12

Suppose a cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 240 meters apart and 40 meters above the roadway as shown in the figure Given below. The cable touches the roadway midway between the towers. Find an equation for the Parabolic shape of cable.

Accepted Answer

A) $y = \frac { 1 } { 300 } x ^ { 2 }$ 
B) $y = \frac { 1 } { 360 } x ^ { 2 }$ $\mathrm { C }$ 
C) $y = \frac { 1 } { 180 } x ^ { 2 }$ 
D)  $y = 360 x ^ { 2 }$ 
E)  $y = 180 x ^ { 2 }$ 
A) $y = \frac { 1 } { 300 } x ^ { 2 }$ 
B) $y = \frac { 1 } { 360 } x ^ { 2 }$ $\mathrm { C }$ 
C) $y = \frac { 1 } { 180 } x ^ { 2 }$ 
D)  $y = 360 x ^ { 2 }$ 
E)  $y = 180 x ^ { 2 }$

Question 13

Find a set of parametric equations for the rectangular equation $y = 4 x ^ { 2 } \text { that satisfies }$ the condition $t = 2$ at the point $( 2,16 )$.

Accepted Answer

A)  $x = t + 2 , y = 8 t + 16$ 
B)  $x = t , y = 8 t$ 
C)  $x = t + 2 , y = 4 t ^ { 2 } + 16 t + 16$ 
D)  $x = t , y = 4 t$ 
E)  $x = t , y = 4 t ^ { 2 }$ 
A)  $x = t + 2 , y = 8 t + 16$ 
B)  $x = t , y = 8 t$ 
C)  $x = t + 2 , y = 4 t ^ { 2 } + 16 t + 16$ 
D)  $x = t , y = 4 t$ 
E)  $x = t , y = 4 t ^ { 2 }$

Question 14

Find the area of the surface generated by revolving the curve $x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }$ $y$-axis on the interval $3 \leq t \leq 4$. Round your answer to two decimal places.

Accepted Answer

A)  $1427.12$ 
B)  $1178.96$ 
C)  $1049.37$ 
D)  $589.48$ 
E)  $1898.04$ 
A)  $1427.12$ 
B)  $1178.96$ 
C)  $1049.37$ 
D)  $589.48$ 
E)  $1898.04$

Question 15

The parametric equations for the path of a projectile launched at a height h feet above the ground, at an angle $\theta$ with the horizontal and having an initial velocity of $v _ { 0 }$ feet per second is given by $x = \left( v _ { 0 } \cos \theta \right) t$ and $y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$. The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of $\theta$ degrees with the horizontal at a speed of 100 miles per hour as shown in the figure. Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run using the parametric equations $x = \left( \frac { 440 } { 3 } \cos \theta \right) t$ and $y = 3 + \left( \frac { 440 } { 3 } \sin \theta \right) t - 16 t ^ { 2 }$. Round your answer to one decimal place.

Accepted Answer

A)  $25.5 ^ { \circ }$ 
B)  $19.4 ^ { \circ }$ 
C)  $3.4 ^ { \circ }$ 
D)  $4.5 ^ { \circ }$ 
E)  $11.3 ^ { \circ }$ 
A)  $25.5 ^ { \circ }$ 
B)  $19.4 ^ { \circ }$ 
C)  $3.4 ^ { \circ }$ 
D)  $4.5 ^ { \circ }$ 
E)  $11.3 ^ { \circ }$

Question 16

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 \cos \theta , y = 4 \sin \theta \text {. }$

Accepted Answer

A)  $- \frac { \tan ^ { 3 } \theta } { 4 }$ 
B)  $- \frac { \csc ^ { 3 } \theta } { 4 }$ 
C)  $\frac { \csc ^ { 3 } \theta } { 4 }$ 
D)  $- \frac { \sec ^ { 3 } \theta } { 4 }$ 
E)  $\frac { \sec ^ { 3 } \theta } { 4 }$ 
A)  $- \frac { \tan ^ { 3 } \theta } { 4 }$ 
B)  $- \frac { \csc ^ { 3 } \theta } { 4 }$ 
C)  $\frac { \csc ^ { 3 } \theta } { 4 }$ 
D)  $- \frac { \sec ^ { 3 } \theta } { 4 }$ 
E)  $\frac { \sec ^ { 3 } \theta } { 4 }$

Question 17

Match the equation with its graph. $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 18

$\text { Find an equation of the parabola with vertex } ( 0,8 ) \text { and directrix } y = - 3 \text {. }$

Accepted Answer

A)  $y ^ { 2 } = 44 ( x - 8 )$ 
B)  $x ^ { 2 } = 44 ( y - 8 )$ 
C)  $x ^ { 2 } = 44 ( y + 8 )$ 
D)  $x ^ { 2 } = 22 ( y - 8 )$ 
E)  $y ^ { 2 } = 22 ( x - 8 )$ 
A)  $y ^ { 2 } = 44 ( x - 8 )$ 
B)  $x ^ { 2 } = 44 ( y - 8 )$ 
C)  $x ^ { 2 } = 44 ( y + 8 )$ 
D)  $x ^ { 2 } = 22 ( y - 8 )$ 
E)  $y ^ { 2 } = 22 ( x - 8 )$

Question 19

Find  the area of the surface generated by revolving the curve $x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }$ $y$-axis on the interval $3 \leq t \leq 4$. Round your answer to two decimal places.

Accepted Answer

A)  $1427.12$ 
B)  $1178.96$ 
C)  $1049.37$ 
D)  $589.48$ 
E)  $1898.04$ 
A)  $1427.12$ 
B)  $1178.96$ 
C)  $1049.37$ 
D)  $589.48$ 
E)  $1898.04$

Question 20

Suppose a church window is bounded above by a parabola and below by the arc of a circle as shown in figure given below. Find the surface area of the window. Round your answer to Three decimal places.

Accepted Answer

A)  $147.477 \mathrm { ft } ^ { 2 }$ 
B)  $62.143 \mathrm { ft } ^ { 2 }$ 
C)  $81.292 \mathrm { ft } ^ { 2 }$ 
D)  $53.610 \mathrm { ft } ^ { 2 }$ 
E)  $184.590 \mathrm { ft } ^ { 2 }$ 
A)  $147.477 \mathrm { ft } ^ { 2 }$ 
B)  $62.143 \mathrm { ft } ^ { 2 }$ 
C)  $81.292 \mathrm { ft } ^ { 2 }$ 
D)  $53.610 \mathrm { ft } ^ { 2 }$ 
E)  $184.590 \mathrm { ft } ^ { 2 }$

Write the corresponding rectangular equation for the curve represented by the parametric equations $x = 9 \cos \theta , y = 9 \sin \theta$ by eliminating the parameter.

Find a polar equation for the parabola with its focus at the pole and vertex $\left( 3 , - \frac { 1 } { 2 } \pi \right)$

Find an equation of the ellipse with vertices $( 0,8 ) , ( 14,8 ) \text { and eccentricity } e = \frac { 2 } { 7 }$

Find the area of the common interior of the polar equations $r = 4 ( 1 + \sin \theta )$ and $r = 4 ( 1 - \sin \theta )$ by sketching the graph of the equations using the graphing utility.

Find the area of the surface generated by revolving the curve about the given axis. $x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2$ (i) $x$ -axis;(ii) $y$ -axis

Find the eccentricity of the polar equation $r = - \frac { 40 } { 20 + 41 \sin \theta }$

Find the center, foci, vertices, and eccentricity of the ellipse. $\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 9 } = 1$

Find the eccentricity of the polar equation $r = \frac { 78 } { 26 + \cos \theta }$

Find a polar equation for the parabola with its focus at the pole, eccentricity e = 1, and directrix $x = - 8$ .

Find the distance from the pole to the directrix for the conic $r ( 16 + \sin \theta ) = 32$

Find a set of parametric equations for the rectangular equation $y = 4 x ^ { 2 } \text { that satisfies }$ the condition $t = 2$ at the point $( 2,16 )$ .

Find the area of the surface generated by revolving the curve $x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }$ $y$ -axis on the interval $3 \leq t \leq 4$ . Round your answer to two decimal places.

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 \cos \theta , y = 4 \sin \theta \text {. }$

Match the equation with its graph. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$

$\text { Find an equation of the parabola with vertex } ( 0,8 ) \text { and directrix } y = - 3 \text {. }$

Find the area of the surface generated by revolving the curve $x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }$ $y$ -axis on the interval $3 \leq t \leq 4$ . Round your answer to two decimal places.

Suppose a church window is bounded above by a parabola and below by the arc of a circle as shown in figure given below. Find the surface area of the window. Round your answer to Three decimal places.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 10: Conics and Calculus

Write the corresponding rectangular equation for the curve represented by the parametric equations x=9cos⁡θ,y=9sin⁡θx = 9 \cos \theta , y = 9 \sin \thetax=9cosθ,y=9sinθ by eliminating the parameter.

Find a polar equation for the parabola with its focus at the pole and vertex (3,−12π)\left( 3 , - \frac { 1 } { 2 } \pi \right)(3,−21​π)

Find an equation of the ellipse with vertices (0,8),(14,8) and eccentricity e=27( 0,8 ) , ( 14,8 ) \text { and eccentricity } e = \frac { 2 } { 7 }(0,8),(14,8) and eccentricity e=72​

Find the area of the common interior of the polar equations r=4(1+sin⁡θ)r = 4 ( 1 + \sin \theta )r=4(1+sinθ) and r=4(1−sin⁡θ)r = 4 ( 1 - \sin \theta )r=4(1−sinθ) by sketching the graph of the equations using the graphing utility.

Find the area of the surface generated by revolving the curve about the given axis. x=3cos⁡3θ,y=3sin⁡3θ,0≤θ≤π/2x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2x=3cos3θ,y=3sin3θ,0≤θ≤π/2 (i) xxx -axis;(ii) yyy -axis

Find the eccentricity of the polar equation r=−4020+41sin⁡θr = - \frac { 40 } { 20 + 41 \sin \theta }r=−20+41sinθ40​

Find the center, foci, vertices, and eccentricity of the ellipse. (x−3)225+(y+2)29=1\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 9 } = 125(x−3)2​+9(y+2)2​=1

Find the eccentricity of the polar equation r=7826+cos⁡θr = \frac { 78 } { 26 + \cos \theta }r=26+cosθ78​

Find a polar equation for the parabola with its focus at the pole, eccentricity e = 1, and directrix x=−8x = - 8x=−8 .

Find the distance from the pole to the directrix for the conic r(16+sin⁡θ)=32r ( 16 + \sin \theta ) = 32r(16+sinθ)=32

Find a set of parametric equations for the rectangular equation y=4x2 that satisfies y = 4 x ^ { 2 } \text { that satisfies }y=4x2 that satisfies the condition t=2t = 2t=2 at the point (2,16)( 2,16 )(2,16) .

Find the area of the surface generated by revolving the curve x=13t3,t+9 about the x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }x=31​t3,t+9 about the yyy -axis on the interval 3≤t≤43 \leq t \leq 43≤t≤4 . Round your answer to two decimal places.

Match the equation with its graph. x29+y24=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 19x2​+4y2​=1

Find an equation of the parabola with vertex (0,8) and directrix y=−3. \text { Find an equation of the parabola with vertex } ( 0,8 ) \text { and directrix } y = - 3 \text {. } Find an equation of the parabola with vertex (0,8) and directrix y=−3.

Find the area of the surface generated by revolving the curve x=13t3,t+9 about the x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }x=31​t3,t+9 about the yyy -axis on the interval 3≤t≤43 \leq t \leq 43≤t≤4 . Round your answer to two decimal places.

Suppose a church window is bounded above by a parabola and below by the arc of a circle as shown in figure given below. Find the surface area of the window. Round your answer to Three decimal places.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Write the corresponding rectangular equation for the curve represented by the parametric equations $x = 9 \cos \theta , y = 9 \sin \theta$ by eliminating the parameter.

Find a polar equation for the parabola with its focus at the pole and vertex $\left( 3 , - \frac { 1 } { 2 } \pi \right)$

Find an equation of the ellipse with vertices $( 0,8 ) , ( 14,8 ) \text { and eccentricity } e = \frac { 2 } { 7 }$

Find the area of the common interior of the polar equations $r = 4 ( 1 + \sin \theta )$ and $r = 4 ( 1 - \sin \theta )$ by sketching the graph of the equations using the graphing utility.

Find the area of the surface generated by revolving the curve about the given axis. $x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2$ (i) $x$ -axis;(ii) $y$ -axis

Find the eccentricity of the polar equation $r = - \frac { 40 } { 20 + 41 \sin \theta }$

Find the center, foci, vertices, and eccentricity of the ellipse. $\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { ( y + 2 ) ^ { 2 } } { 9 } = 1$

Find the eccentricity of the polar equation $r = \frac { 78 } { 26 + \cos \theta }$

Find a polar equation for the parabola with its focus at the pole, eccentricity e = 1, and directrix $x = - 8$ .

Find the distance from the pole to the directrix for the conic $r ( 16 + \sin \theta ) = 32$

Find a set of parametric equations for the rectangular equation $y = 4 x ^ { 2 } \text { that satisfies }$ the condition $t = 2$ at the point $( 2,16 )$ .

Find the area of the surface generated by revolving the curve $x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }$ $y$ -axis on the interval $3 \leq t \leq 4$ . Round your answer to two decimal places.

Match the equation with its graph. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$

$\text { Find an equation of the parabola with vertex } ( 0,8 ) \text { and directrix } y = - 3 \text {. }$

Find the area of the surface generated by revolving the curve $x = \frac { 1 } { 3 } t ^ { 3 } , t + 9 \text { about the }$ $y$ -axis on the interval $3 \leq t \leq 4$ . Round your answer to two decimal places.