Exam 9: Parametric Equations Polar Coordinates and Conic Sections

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$

(Multiple Choice)

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Question 21

Describe the conic: $x ^ { 2 } + 4 x + y ^ { 2 } - 8 y - 24 = 0$

(Essay)

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Question 22

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$

(Multiple Choice)

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Question 23

Find a definite integral expression that represents the area of the region inside $r = 1 + \cos \theta$ Then find the exact value of the integral.

(Essay)

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Question 24

Convert the equation $\theta = \frac { \pi } { 3 }$ to rectangular coordinates.

(Multiple Choice)

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Question 25

Convert the equation y = 4 to polar coordinates.

(Multiple Choice)

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Question 26

Find the polar coordinates of the point (-3, 0) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

(Multiple Choice)

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Question 27

Find all the points of intersection of the curves, for $0 \leq \theta < 2 \pi$ : $r = \sin \theta \text { and } r = \cos \theta$

(Essay)

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Question 28

Convert the equation $r = 5 \sec \theta$ to rectangular coordinates.

(Multiple Choice)

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Question 29

Find the rectangular coordinates for the following points whose polar coordinates are given: A .(3, $\pi$ /6) B. (-3, $\pi$ /2) C. (3, 3 $\pi$ /2) D. (3, - $\pi$ /6)

(Essay)

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Question 30

Write the equation of the ellipse with foci $( 0 , \pm 3 )$ and length of minor axis 8.

(Essay)

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Question 31

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 2 \sin 2 \theta \text { and } r = 1$

(Essay)

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Question 32

Find the equation of the tangent line to the parametric curve: $x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$

(Multiple Choice)

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Question 33

Find the polar coordinates of the point $( - 2 \sqrt { 3 } , 2 )$ given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

(Multiple Choice)

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Question 34

Find the length of the parametric curve: $x = e ^ { 2 t } \cos t , \quad y = e ^ { 2 t } \sin t , \quad \mathrm { t } \in [ 0,1 ]$

(Multiple Choice)

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Question 35

Write the equation of the ellipse with foci $( \pm 2,0 )$ and length of major axis 6.

(Essay)

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Question 36

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$

(Multiple Choice)

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Question 37

Find a definite integral expression that represents the area of the region inside one loop of the lemniscate $r ^ { 2 } = \sin 2 \theta$ Then find the exact value of the integral.

(Essay)

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Question 38

Find a definite integral expression that represents the area of the region inside both the cardioid $r = 1 + \sin \theta$ and outside the cardioid $r = 1 - \sin \theta$ Then find the exact value of the integral.

(Essay)

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Question 39

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 1 - \sin \theta \text { and } r = 1 + \cos \theta$

(Essay)

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Question 40

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$

Describe the conic: $x ^ { 2 } + 4 x + y ^ { 2 } - 8 y - 24 = 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$

Find a definite integral expression that represents the area of the region inside $r = 1 + \cos \theta$ Then find the exact value of the integral.

Convert the equation $\theta = \frac { \pi } { 3 }$ to rectangular coordinates.

Convert the equation y = 4 to polar coordinates.

Find the polar coordinates of the point (-3, 0) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

Find all the points of intersection of the curves, for $0 \leq \theta < 2 \pi$ : $r = \sin \theta \text { and } r = \cos \theta$

Convert the equation $r = 5 \sec \theta$ to rectangular coordinates.

Find the rectangular coordinates for the following points whose polar coordinates are given: A .(3, $\pi$ /6) B. (-3, $\pi$ /2) C. (3, 3 $\pi$ /2) D. (3, - $\pi$ /6)

Write the equation of the ellipse with foci $( 0 , \pm 3 )$ and length of minor axis 8.

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 2 \sin 2 \theta \text { and } r = 1$

Find the equation of the tangent line to the parametric curve: $x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$

Find the polar coordinates of the point $( - 2 \sqrt { 3 } , 2 )$ given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

Find the length of the parametric curve: $x = e ^ { 2 t } \cos t , \quad y = e ^ { 2 t } \sin t , \quad \mathrm { t } \in [ 0,1 ]$

Write the equation of the ellipse with foci $( \pm 2,0 )$ and length of major axis 6.

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$

Find a definite integral expression that represents the area of the region inside one loop of the lemniscate $r ^ { 2 } = \sin 2 \theta$ Then find the exact value of the integral.

Find a definite integral expression that represents the area of the region inside both the cardioid $r = 1 + \sin \theta$ and outside the cardioid $r = 1 - \sin \theta$ Then find the exact value of the integral.

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 1 - \sin \theta \text { and } r = 1 + \cos \theta$

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

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Exam 9: Parametric Equations Polar Coordinates and Conic Sections

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=2sin⁡t,y=2cos⁡t,t∈[0,2π]x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]x=2sint,y=2cost,t∈[0,2π]

Describe the conic: x2+4x+y2−8y−24=0x ^ { 2 } + 4 x + y ^ { 2 } - 8 y - 24 = 0x2+4x+y2−8y−24=0

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=cos⁡t,y=cos⁡2t,t∈[0,π]x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]x=cost,y=cos2t,t∈[0,π]

Find a definite integral expression that represents the area of the region inside r=1+cos⁡θr = 1 + \cos \thetar=1+cosθ Then find the exact value of the integral.

Convert the equation θ=π3\theta = \frac { \pi } { 3 }θ=3π​ to rectangular coordinates.

Convert the equation y = 4 to polar coordinates.

Find the polar coordinates of the point (-3, 0) given in rectangular coordinates for θ∈[0,2π]\theta \in [ 0,2 \pi ]θ∈[0,2π] and r≥0r \geq 0r≥0

Find all the points of intersection of the curves, for 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π : r=sin⁡θ and r=cos⁡θr = \sin \theta \text { and } r = \cos \thetar=sinθ and r=cosθ

Convert the equation r=5sec⁡θr = 5 \sec \thetar=5secθ to rectangular coordinates.

Find the rectangular coordinates for the following points whose polar coordinates are given: A .(3, π\piπ /6) B. (-3, π\piπ /2) C. (3, 3 π\piπ /2) D. (3, - π\piπ /6)

Write the equation of the ellipse with foci (0,±3)( 0 , \pm 3 )(0,±3) and length of minor axis 8.

Find all the points of intersection of the curves for 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π : r=2sin⁡2θ and r=1r = 2 \sin 2 \theta \text { and } r = 1r=2sin2θ and r=1

Find the equation of the tangent line to the parametric curve: x=t+1,y=e2t, at t=0x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0x=t+1,y=e2t, at t=0

Find the polar coordinates of the point (−23,2)( - 2 \sqrt { 3 } , 2 )(−23​,2) given in rectangular coordinates for θ∈[0,2π]\theta \in [ 0,2 \pi ]θ∈[0,2π] and r≥0r \geq 0r≥0

Find the length of the parametric curve: x=e2tcos⁡t,y=e2tsin⁡t,t∈[0,1]x = e ^ { 2 t } \cos t , \quad y = e ^ { 2 t } \sin t , \quad \mathrm { t } \in [ 0,1 ]x=e2tcost,y=e2tsint,t∈[0,1]

Write the equation of the ellipse with foci (±2,0)( \pm 2,0 )(±2,0) and length of major axis 6.

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=t−1,y=2t+5,t∈(−∞,∞)x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )x=t−1,y=2t+5,t∈(−∞,∞)

Find a definite integral expression that represents the area of the region inside one loop of the lemniscate r2=sin⁡2θr ^ { 2 } = \sin 2 \thetar2=sin2θ Then find the exact value of the integral.

Find a definite integral expression that represents the area of the region inside both the cardioid r=1+sin⁡θr = 1 + \sin \thetar=1+sinθ and outside the cardioid r=1−sin⁡θr = 1 - \sin \thetar=1−sinθ Then find the exact value of the integral.

Find all the points of intersection of the curves for 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π : r=1−sin⁡θ and r=1+cos⁡θr = 1 - \sin \theta \text { and } r = 1 + \cos \thetar=1−sinθ and r=1+cosθ

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 2 \sin t , \quad y = 2 \cos t , \quad t \in [ 0,2 \pi ]$

Describe the conic: $x ^ { 2 } + 4 x + y ^ { 2 } - 8 y - 24 = 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = \cos t , \quad y = \cos 2 t , \quad t \in [ 0 , \pi ]$

Find a definite integral expression that represents the area of the region inside $r = 1 + \cos \theta$ Then find the exact value of the integral.

Convert the equation $\theta = \frac { \pi } { 3 }$ to rectangular coordinates.

Find the polar coordinates of the point (-3, 0) given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

Find all the points of intersection of the curves, for $0 \leq \theta < 2 \pi$ : $r = \sin \theta \text { and } r = \cos \theta$

Convert the equation $r = 5 \sec \theta$ to rectangular coordinates.

Find the rectangular coordinates for the following points whose polar coordinates are given: A .(3, $\pi$ /6) B. (-3, $\pi$ /2) C. (3, 3 $\pi$ /2) D. (3, - $\pi$ /6)

Write the equation of the ellipse with foci $( 0 , \pm 3 )$ and length of minor axis 8.

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 2 \sin 2 \theta \text { and } r = 1$

Find the equation of the tangent line to the parametric curve: $x = t + 1 , \quad y = e ^ { 2 t } , \quad \text { at } t = 0$

Find the polar coordinates of the point $( - 2 \sqrt { 3 } , 2 )$ given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$

Find the length of the parametric curve: $x = e ^ { 2 t } \cos t , \quad y = e ^ { 2 t } \sin t , \quad \mathrm { t } \in [ 0,1 ]$

Write the equation of the ellipse with foci $( \pm 2,0 )$ and length of major axis 6.

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = t - 1 , \quad y = 2 t + 5 , \quad t \in ( - \infty , \infty )$

Find a definite integral expression that represents the area of the region inside one loop of the lemniscate $r ^ { 2 } = \sin 2 \theta$ Then find the exact value of the integral.

Find a definite integral expression that represents the area of the region inside both the cardioid $r = 1 + \sin \theta$ and outside the cardioid $r = 1 - \sin \theta$ Then find the exact value of the integral.

Find all the points of intersection of the curves for $0 \leq \theta < 2 \pi$ : $r = 1 - \sin \theta \text { and } r = 1 + \cos \theta$