Deck 12: Parametric and Polar Curves

Question 1

Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the
particle and the direction of motion.

-x = 2 cos t, y = 2 sin t, $\pi$$\le$ t $\le$ 2 $\pi$

A)
B)
C)
D)

Answer 1

Question 2

Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the
particle and the direction of motion.

-x = 4 sin t, y = 3 cos t, 0 $\le$ t $\le$ 2 $\pi$

A)
B)
C)
D)

Answer 2

Question 3

Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the
particle and the direction of motion.

-x = 36 , y = 6t, - $\infty$ $\le$ t $\le$ $\infty$

A)
B)
C)
D)

Answer 3

Question 4

Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the
particle and the direction of motion.

-x = 3t + 4, y = 9t + 3, - $\infty$ $\le$ t $\le$ $\infty$

A)
B)
C)
D)

Question 5

Find a parametrization for the curve.

-The line segment with endpoints ( -8, -7) and ( 0, -12)

Question 6

Find a parametrization for the curve.
-The upper half of the parabola x + 3 =

Question 7

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 5(t - sin t), y= 5(1 - cos t), 0 $\le$ t $\le$ 4 $\pi$

A)
B)
C)
D)

Question 8

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = t - sin t, y = 1 - cos t, -4 $\pi$ $\le$ t $\le$ 4 $\pi$

A)
B)
C)
D)

Question 9

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 6t - 6 sin t, y = 6 - 6 cos t, -2 $\pi$ $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 10

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = cos 4t, y = sin 3t, 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 11

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 3 cos t + cos 3t, y = 3 sin t - sin 3t, 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 12

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 6 cos t + 2 cos 3t, y = 6 sin t - 2 sin3t, 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 13

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = cos t + 4 cos 3t, y = 5 cos t - 4 sin 3t, 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 14

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 5 cos 2t + 4 cos 6t, y = 5 sin 2t - 4 sin 6t, 0 $\le$ t $\le$$\pi$

A)

B)

C)

D)

Question 15

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 6 sin t - sin 6t, y = 6 cos t - cos 6t, 0 $\le$ t $\le$ 2 $\pi$

A)

B)

C)

D)

Question 16

Graph the pair of parametric equations with the aid of a graphing calculator.

-x = 3 tan t, y = 4 sec t; - $\pi$ /2 $\le$ t $\le$$\pi$ /2

A)

B)

C)

D)

Question 17

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = 11 i + 2 j + 10 k, 1 $\le$ t $\le$ 2

A) 1575
B) 525
C) 105
D) 35

Question 18

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = ( 3 - 6t)i + ( 2 + 2t)j + ( 9t - 7)k, -5 $\le$ t $\le$ 3

A) -22
B) 968
C) -242
D) 88

Question 19

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = ( 7 + 2 $t^{2}$ )i + (2 $t^{2}$ - 3)j + ( 10 - $t^{2}$ )k, 1 $\le$ t $\le$ 4

A) 51
B) 135
C) 153
D) 45

Question 20

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = 3ti + j + k; -8 $\le$ t $\le$ -7

A) -375
B) 25
C) -75
D) 5

Question 21

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = ( 7 2t)j + ( 7 2t)k; $\pi$ $\le$ t $\le$ $\pi$

A) 0
B)
C)
D) 21

Question 22

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = i + ( 7t sin t)j + ( 7t cos t)k ; -4 $\le$ t $\le$ 5

A) 294
B) 126
C) 94.5
D) 150.5

Question 23

Calculate the arc length of the indicated portion of the curve r(t).

-r(t) = ( 7t sin t + 7 cos t)i + ( 7 t cos t - 7 sin t)j ; -2 $\le$ t $\le$ 5

A) 73.5
B) 101.5
C) 203
D) 147

Question 24

Plot the point whose polar coordinates are given.

-( 2, 9 $\pi$ /4)

A)

B)

C)

D)

Question 25

Plot the point whose polar coordinates are given.

-( 4, $\pi$ /2)

A)

B)

C)

D)

Question 26

Plot the point whose polar coordinates are given.

-( -4, 0)

A)

B)

C)

D)

Question 27

Plot the point whose polar coordinates are given.

-( -4, 2 $\pi$ /3)

A)

B)

C)

D)

Question 28

Determine if the given polar coordinates represent the same point.

-( 10, $\pi$ /4), ( 10, 5 $\pi$ /4)

Question 29

Determine if the given polar coordinates represent the same point.

-( 2, $\pi$ /4), (-2, $\pi$ /4)

Question 30

Determine if the given polar coordinates represent the same point.

-( 6, $\pi$ /4), (-6, - $\pi$ /4)

Question 31

Determine if the given polar coordinates represent the same point.

-( 2, $\pi$ /2), (-2, 3 $\pi$ /2)

Question 32

Determine if the given polar coordinates represent the same point.

-( 2, $\pi$ /3), (-2, -2 $\pi$ /3)

Question 33

Determine if the given polar coordinates represent the same point.

-( 10, $\pi$ /6), ( 10, 7 $\pi$ /6)

Question 34

Determine if the given polar coordinates represent the same point.

-( 8, $\pi$ /6), (-8, 7 $\pi$ /6)

Question 35

Determine if the given polar coordinates represent the same point.

-( 3, -4 $\pi$ /3), (-3, 5 $\pi$ /3)

Question 36

Determine if the given polar coordinates represent the same point.

-(r, $\theta$ ), (-r, $\theta$ + $\pi$ )

Question 37

Find the Cartesian coordinates of the given point.

-

A) ( 4, 0)
B) (0, 4)
C) (-4, 0)
D) ( 0, -4)

Question 38

Find the Cartesian coordinates of the given point.

-( , $\pi$ /6)

A)
B)
C)
D)

Question 39

Find the Cartesian coordinates of the given point.

-(-4, - $\pi$ /3)

A) (-2 , 2 )
B)(2, 2 )
C) (-2 , 2 )
D) (-2 , -2)

Question 40

Find the Cartesian coordinates of the given point.

-( -5, $\pi$ )

A) ( 0, 5)
B) (-5, 0)
C) (5, 0)
D) ( 0, -5)

Question 41

Find the Cartesian coordinates of the given point.

-( -8, 0)

A) ( 8, 0)
B) (-8, 0)
C) ( 0, -8)
D) (0, 8)

Question 42

Find the Cartesian coordinates of the given point.

-

A) ( 5 , 5)
B)( 10 , 10 )
C) ( 5 , 5 )
D) ( 10, 10)

Question 43

Find the Cartesian coordinates of the given point.

-

A)
B)
C)
D)

Question 44

Find the Cartesian coordinates of the given point.

-

A)
B)
C)
D)

Question 45

Find the Cartesian coordinates of the given point.

-

A) ( 4 , -4 )
B) ( 4 , -4)
C) ( -4 , -4 )
D) (-8 , -8 )

Question 46

Find the Cartesian coordinates of the given point.

-

A) ( -4 , -4 )
B) ( -4 , -4 )
C) ( 4 , -4 )
D) ( 4 , -4 )

Question 47

Replace the polar equation with an equivalent Cartesian equation.

-r cos $\theta$ = 13

A) 13y = 1
B) x = 13
C) y = 13
D) 13x = 1

Question 48

Replace the polar equation with an equivalent Cartesian equation.

-r = -12 csc $\theta$

A) -12x = 1
B) y = -12
C) x= -12
D) -12y = 1

Question 49

Replace the polar equation with an equivalent Cartesian equation.

-6r cos $\theta$ + 2r sin $\theta$ = 1

A) 6x + 2y = +
B) + = 1
C) 6y + 2x = 1
D) 6x + 2y = 1

Question 50

Replace the polar equation with an equivalent Cartesian equation.

-r =

A) + = 1
B) 7y - 8x = 1
C) 7x - 8y = 1
D) = 1

Question 51

Replace the polar equation with an equivalent Cartesian equation.

-r = 5 cot $\theta$ csc $\theta$

A)y = 5
B) y =
C) = 5x
D) y = 5x

Question 52

Replace the polar equation with an equivalent Cartesian equation.

- sin 2 $\theta$ = 40

A) y = 20x
B) y =
C) y = 40
D) y =

Question 53

Replace the polar equation with an equivalent Cartesian equation.

- + 2 sin $\theta$ cos $\theta$ = 625

A) + 3 = 625
B) x + y =25
C) + = 625
D) x + y = ±25

Question 54

Replace the polar equation with an equivalent Cartesian equation.

-r = 26 sin $\theta$

A) y = 26
B) + = 169
C) + = 169
D) + = 169

Question 55

Replace the polar equation with an equivalent Cartesian equation.

- = 30r cos $\theta$

A) x = 30
B) + = 225
C) + = 225
D) + = 0

Question 56

Replace the polar equation with an equivalent Cartesian equation.

- = 46r cos $\theta$ - 6r sin $\theta$ - 9

A) 46x - 6y = 9
B) + = 9
C) + = 9
D) + = 529

Question 57

Graph the polar equation.

-r = 3( 1 + 2 sin $\theta$ )

A)

B)

C)

D)

Question 58

Graph the polar equation.

-r = 6 - cos $\theta$

A)

B)

C)

D)

Question 59

Graph the polar equation.

-r = 8 sin 4 $\theta$

A)

B)

C)

D)

Question 60

Graph the polar equation.

-r = 4 cos 8 $\theta$

A)

B)

C)

D)

Question 61

Graph the polar equation.

- = 60 cos 3 $\theta$

A)

B)

C)

D)

Question 62

Graph the polar equation.

- = 18 sin 3 $\theta$

A)

B)

C)

D)

Question 63

Graph the polar equation.

-r = 4(1 -sin $\theta$ )

A)

B)

C)

D)

Question 64

Graph the polar equation.

-r = 3 $\theta$

A)

B)

C)

D)

Question 65

Graph the polar equation.

-r = - sin $\theta$

A)

B)

C)

D)

Question 66

Graph the polar equation.

-r = 2 + cos $\theta$

A)

B)

C)

D)

Question 67

Find the area of the specified region.

-Inside the limacon r = 9 + 6 sin $\theta$

A) 99 $\pi$
B) 198 $\pi$ - 108
C) 99 $\pi$ + 108
D) 198 $\pi$

Question 68

Find the area of the specified region.

-Inside the cardioid r = $\alpha$ (1 + sin $\theta$ ), $\alpha$ > 0

A) - 2
B)
C) + 2
D) 2 $\alpha$ $\pi$

Question 69

Find the area of the specified region.

-Inside one leaf of the four-leaved rose r = 7 sin 2 $\theta$

A)
B) -
C)
D)

Question 70

Find the area of the specified region.

-Inside the lemniscate = sin 2 $\theta$ , a > 0

A) 0
B)
C) $\pi$
D)

Question 71

Find the area of the specified region.

-Inside one loop of the lemniscate = 6 cos 2 $\theta$

A) 6
B)
C) 12
D) 3

Question 72

Find the area of the specified region.

-Inside the six-leaved rose = 8 cos 3 $\theta$

A) 4
B) 8
C) 16
D)

Question 73

Find the area of the specified region.

-Inside the circle r = 2 cos $\theta$ + 5 sin $\theta$

A) + 5
B) 5 $\pi$
C)
D)

Question 74

Find the area of the specified region.

-Inside the smaller loop of the limacon r = 5 + 10 sin $\theta$

A) 25 $\pi$
B)
C) 5(3 - $\pi$ )
D) (2 $\pi$ - 3 )

Question 75

Find the area of the specified region.

-Inside the graph of r = 3 + cos 3 $\theta$

A)
B) 19 $\pi$
C) +
D) 19 $\pi$ +

Question 76

Find the area of the specified region.

-Inside the three-leaved rose r = 6 cos 3 $\theta$

A) $\pi$
B) 3 $\pi$
C) 18 $\pi$
D) 9 $\pi$

Question 77

Find the area of the specified region.

-Inside the circle r = sin $\theta$ and outside the cardioid r = 1 + cos $\theta$

A)
B)
C)
D)

Question 78

Find the area of the specified region.

-Shared by the circles r = 3 cos $\theta$ and r = 3 sin $\theta$

A) ( $\pi$ - 2)
B) (4 $\pi$ - 3 )
C) (2 - )
D) $\pi$

Question 79

Find the area of the specified region.

-Shared by the circles r = 3 and r = 6 sin $\theta$

A) $\pi$
B) (2 $\pi$ + 3 )
C) (4 $\pi$ - 3 )
D) 3 $\pi$

Question 80

Find the area of the specified region.

-Shared by the circle r = 8 and the cardioid r = 8(1 + sin $\theta$ )

A) 16(5 $\pi$ - 8)
B) 64
C) 0
D) 16(5 $\pi$ +8)