Question 1

The rectangular equation of the conic section $r = \frac { 3 } { 5 + 4 \cos \theta }$ is

Accepted Answer

A)  $25 x ^ { 2 } + 20 y = 3$ 
B)  $9 x ^ { 2 } + 25 y ^ { 2 } - 24 x = 9$ 
C)  $9 x ^ { 2 } + 25 y ^ { 2 } + 24 y = 9$ 
D)  $9 x ^ { 2 } + 25 y ^ { 2 } + 24 x = 9$ 
E)  $9 x ^ { 2 } + 25 y ^ { 2 } - 24 y = 9$ 
A)  $25 x ^ { 2 } + 20 y = 3$ 
B)  $9 x ^ { 2 } + 25 y ^ { 2 } - 24 x = 9$ 
C)  $9 x ^ { 2 } + 25 y ^ { 2 } + 24 y = 9$ 
D)  $9 x ^ { 2 } + 25 y ^ { 2 } + 24 x = 9$ 
E)  $9 x ^ { 2 } + 25 y ^ { 2 } - 24 y = 9$

Question 2

The length of the curve $x ( t ) = 9 - t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

Accepted Answer

A)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$ 
B)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$ 
C)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$ 
D)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$ 
E)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$ 
A)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$ 
B)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$ 
C)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$ 
D)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$ 
E)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$

Question 3

The length of $r = 6 \cos \theta$ with $x \in \left[ 0 , \frac { \pi } { 2 } \right]$ is

Accepted Answer

A)  $9 \pi$ 
B)  $6 \pi$ 
C)  $3 \pi$ 
D)  $\frac { 3 \pi } { 2 }$ 
E)  $\frac { 3 \pi } { 5 }$ 
A)  $9 \pi$ 
B)  $6 \pi$ 
C)  $3 \pi$ 
D)  $\frac { 3 \pi } { 2 }$ 
E)  $\frac { 3 \pi } { 5 }$

Question 4

The area of the surface generated by revolving the curve $x ( t ) = 3 t , \quad y = \sqrt { 7 } t + 1$ with $t \in [ 0,1 ]$ about the x-axis is

Accepted Answer

A)  $256 \pi \ln 2$ 
B)  $128 \pi \ln 2$ 
C)  $64 \pi \ln 2$ 
D)  $32 \pi \ln 2$ 
E)  $16 \pi \ln 2$ 
A)  $256 \pi \ln 2$ 
B)  $128 \pi \ln 2$ 
C)  $64 \pi \ln 2$ 
D)  $32 \pi \ln 2$ 
E)  $16 \pi \ln 2$

Question 5

The conic section $r = \frac { 5 } { 7 + 8 \sin \theta }$ is

Accepted Answer

A) A straight line 
B) A circle 
C) A parabola 
D) An ellipse 
E) A hyperbola 
A) A straight line 
B) A circle 
C) A parabola 
D) An ellipse 
E) A hyperbola

Question 6

If A is the area of the intersection of the regions inside $6 \pi - 16$ and outside $r = \sqrt { 3 } \text {, }$ then A is

Accepted Answer

A)  $6 \pi - 16$ 
B)  $3 \sqrt { 3 } - \pi$ 
C)  $\frac { \pi + 1 } { 2 }$ 
D)  $\frac { \pi - 2 } { 2 }$ 
E)  $\frac { 8 - \pi } { 4 }$ 
A)  $6 \pi - 16$ 
B)  $3 \sqrt { 3 } - \pi$ 
C)  $\frac { \pi + 1 } { 2 }$ 
D)  $\frac { \pi - 2 } { 2 }$ 
E)  $\frac { 8 - \pi } { 4 }$

Question 7

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2 ( 1 - \cos \theta )$ and $r = 2 ( 1 + \cos \theta ),$ then A is

Accepted Answer

A)  $6 \pi - 16$ 
B)  $\frac { \pi } { 2 }$ 
C)  $\frac { \pi + 1 } { 2 }$ 
D)  $\frac { \pi - 2 } { 2 }$ 
E)  $\frac { 8 - \pi } { 4 }$ 
A)  $6 \pi - 16$ 
B)  $\frac { \pi } { 2 }$ 
C)  $\frac { \pi + 1 } { 2 }$ 
D)  $\frac { \pi - 2 } { 2 }$ 
E)  $\frac { 8 - \pi } { 4 }$

Question 8

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } , y ( t ) = 3 e ^ { t }$ with $t \in ( - \infty , \infty )$ , is

Accepted Answer

A)  $y = \frac { 3 } { x }$ 
B)  $y = \frac { x } { 3 }$ 
C)  $y = - \frac { 3 } { x }$ 
D)  $y = 3 x$ 
E)  $y = - \frac { x } { 3 }$ 
A)  $y = \frac { 3 } { x }$ 
B)  $y = \frac { x } { 3 }$ 
C)  $y = - \frac { 3 } { x }$ 
D)  $y = 3 x$ 
E)  $y = - \frac { x } { 3 }$

Question 9

The polar equation that corresponds to the rectangular equation $y = x ^ { 2 }$ is

Accepted Answer

A)  $r = - \tan \theta \sec \theta$ 
B)  $r = \tan \theta \sec \theta$ 
C)  $r = \cot \theta \csc \theta$ 
D)  $r = - \cot \theta \csc \theta$ 
E)  $r ^ { 2 } = \tan \theta \sec \theta$ 
A)  $r = - \tan \theta \sec \theta$ 
B)  $r = \tan \theta \sec \theta$ 
C)  $r = \cot \theta \csc \theta$ 
D)  $r = - \cot \theta \csc \theta$ 
E)  $r ^ { 2 } = \tan \theta \sec \theta$

Question 10

The rectangular equation that corresponds to the polar equation $\frac { r } { 2 } = \theta$ is

Accepted Answer

A)  $y = 2 x$ 
B)  $x ^ { 2 } + y ^ { 2 } = 2 x$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 \tan ^ { - 1 } \frac { x } { y }$ 
D)  $x ^ { 2 } + y ^ { 2 } = - 2 x$ 
E)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 \tan ^ { - 1 } \frac { y } { x }$ 
A)  $y = 2 x$ 
B)  $x ^ { 2 } + y ^ { 2 } = 2 x$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 \tan ^ { - 1 } \frac { x } { y }$ 
D)  $x ^ { 2 } + y ^ { 2 } = - 2 x$ 
E)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 \tan ^ { - 1 } \frac { y } { x }$

Question 11

For a$\neq$ 0, the polar curve $r ^ { 2 } = a \sin 2 \theta$ is a

Accepted Answer

A) Circle 
B) Two-petal rose 
C) Four-petal rose 
D) Limaçon 
E) Lemniscate 
A) Circle 
B) Two-petal rose 
C) Four-petal rose 
D) Limaçon 
E) Lemniscate

Question 12

For a $\neq$ 0, the polar curve $r = a ( 1 + 2 \cos \theta )$ is a

Accepted Answer

A) Straight line 
B) Circle 
C) Three-petal rose 
D) Cardioid 
E) Limaçon 
A) Straight line 
B) Circle 
C) Three-petal rose 
D) Cardioid 
E) Limaçon

Question 13

The area of the surface generated by revolving the curve $x ( t ) = 1 - t ^ { 2 } , y = 3 + 2 t$ with $t \in [ 0,1 ]$ about the x-axis is

Accepted Answer

A)  $4 \pi ( \sqrt { 2 } + 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
B)  $\pi ( \sqrt { 2 } + 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
C)  $2 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
D)  $4 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
E)  $8 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
A)  $4 \pi ( \sqrt { 2 } + 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
B)  $\pi ( \sqrt { 2 } + 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
C)  $2 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
D)  $4 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$ 
E)  $8 \pi ( \sqrt { 2 } - 1 ) \left( 1 + \sinh ^ { - 1 } 1 \right)$

Question 14

If A is the area of the region bounded by one petal of the four-petal rose $r = \cos 2 \theta,$ then A is

Accepted Answer

A)  $\frac { \pi } { 8 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\pi$ 
A)  $\frac { \pi } { 8 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\pi$

Question 15

Find the points on the curve $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ where the tangent line is vertical.

Accepted Answer

A)  $\left( \frac { 27 } { 4 } , - 2 \right)$ 
B)  $( 0,1 ) , \left( \frac { 27 } { 4 } , - 2 \right)$ 
C)  $( 0,1 )$ 
D)  $( 0,1 ) , \left( \frac { 27 } { 4 } , 2 \right)$ 
E)  $( 0,1 ) , \left( - \frac { 27 } { 4 } , 2 \right)$ 
A)  $\left( \frac { 27 } { 4 } , - 2 \right)$ 
B)  $( 0,1 ) , \left( \frac { 27 } { 4 } , - 2 \right)$ 
C)  $( 0,1 )$ 
D)  $( 0,1 ) , \left( \frac { 27 } { 4 } , 2 \right)$ 
E)  $( 0,1 ) , \left( - \frac { 27 } { 4 } , 2 \right)$

Question 16

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2$ and $r = 3 - 2 \cos \theta,$ then A is

Accepted Answer

A)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 15 }$ 
B)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 12 }$ 
C)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 8 }$ 
D)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 6 }$ 
E)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 4 }$ 
A)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 15 }$ 
B)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 12 }$ 
C)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 8 }$ 
D)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 6 }$ 
E)  $\frac { 38 \pi - 33 \sqrt { 3 } } { 4 }$

Question 17

The parametric equations of the line segment from (1,0) to (0,1) with t $\in$[0,1] are

Accepted Answer

A)  $x ( t ) = t , y ( t ) = 1 - t$ 
B)  $x ( t ) = t , y ( t ) = t - 1$ 
C)  $x ( t ) = 1 - t , y ( t ) = t$ 
D)  $x ( t ) = t - 1 , y ( t ) = t$ 
E)  $x ( t ) = t - 1 , y ( t ) = 1 - t$ 
A)  $x ( t ) = t , y ( t ) = 1 - t$ 
B)  $x ( t ) = t , y ( t ) = t - 1$ 
C)  $x ( t ) = 1 - t , y ( t ) = t$ 
D)  $x ( t ) = t - 1 , y ( t ) = t$ 
E)  $x ( t ) = t - 1 , y ( t ) = 1 - t$

Question 18

If A is the area of the region bounded by one leaf of the lemniscates $r ^ { 2 } = 3 \sin 2 \theta,$ then A is

Accepted Answer

A)  $\frac { 9 } { 2 }$ 
B)  $\frac { 7 } { 2 }$ 
C)  $\frac { 5 } { 2 }$ 
D)  $\frac { 3 } { 2 }$ 
E)  $\frac { 1 } { 2 }$ 
A)  $\frac { 9 } { 2 }$ 
B)  $\frac { 7 } { 2 }$ 
C)  $\frac { 5 } { 2 }$ 
D)  $\frac { 3 } { 2 }$ 
E)  $\frac { 1 } { 2 }$

Question 19

The length of the curve $x ( t ) = e ^ { t } \cos t , y ( t ) = e ^ { t } \sin t$ with $t \in [ 0 , \pi ]$ is

Accepted Answer

A)  $\sqrt { 3 } \left( e ^ { \pi } - 1 \right)$ 
B)  $\sqrt { 3 } \left( e ^ { \pi } + 1 \right)$ 
C)  $\sqrt { 5 } \left( e ^ { \pi } - 1 \right)$ 
D)  $\sqrt { 2 } \left( e ^ { \pi } + 1 \right)$ 
E)  $\sqrt { 2 } \left( e ^ { \pi } - 1 \right)$ 
A)  $\sqrt { 3 } \left( e ^ { \pi } - 1 \right)$ 
B)  $\sqrt { 3 } \left( e ^ { \pi } + 1 \right)$ 
C)  $\sqrt { 5 } \left( e ^ { \pi } - 1 \right)$ 
D)  $\sqrt { 2 } \left( e ^ { \pi } + 1 \right)$ 
E)  $\sqrt { 2 } \left( e ^ { \pi } - 1 \right)$

Question 20

Let $x ( t ) = \ln t , y ( t ) = \frac { 1 } { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

Accepted Answer

A)  $- \frac { 2 } { t ^ { 2 } }$ 
B)  $\frac { 1 } { t ^ { 2 } }$ 
C)  $- \frac { 1 } { t }$ 
D)  $\frac { 1 } { t }$ 
E)  $- \frac { 1 } { t ^ { 2 } }$ 
A)  $- \frac { 2 } { t ^ { 2 } }$ 
B)  $\frac { 1 } { t ^ { 2 } }$ 
C)  $- \frac { 1 } { t }$ 
D)  $\frac { 1 } { t }$ 
E)  $- \frac { 1 } { t ^ { 2 } }$

The rectangular equation of the conic section $r = \frac { 3 } { 5 + 4 \cos \theta }$ is

The length of the curve $x ( t ) = 9 - t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

The length of $r = 6 \cos \theta$ with $x \in \left[ 0 , \frac { \pi } { 2 } \right]$ is

The area of the surface generated by revolving the curve $x ( t ) = 3 t , \quad y = \sqrt { 7 } t + 1$ with $t \in [ 0,1 ]$ about the x-axis is

The conic section $r = \frac { 5 } { 7 + 8 \sin \theta }$ is

If A is the area of the intersection of the regions inside $6 \pi - 16$ and outside $r = \sqrt { 3 } \text {, }$ then A is

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2 ( 1 - \cos \theta )$ and $r = 2 ( 1 + \cos \theta ),$ then A is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } , y ( t ) = 3 e ^ { t }$ with $t \in ( - \infty , \infty )$ , is

The polar equation that corresponds to the rectangular equation $y = x ^ { 2 }$ is

The rectangular equation that corresponds to the polar equation $\frac { r } { 2 } = \theta$ is

For a $\neq$ 0, the polar curve $r ^ { 2 } = a \sin 2 \theta$ is a

For a $\neq$ 0, the polar curve $r = a ( 1 + 2 \cos \theta )$ is a

The area of the surface generated by revolving the curve $x ( t ) = 1 - t ^ { 2 } , y = 3 + 2 t$ with $t \in [ 0,1 ]$ about the x-axis is

If A is the area of the region bounded by one petal of the four-petal rose $r = \cos 2 \theta,$ then A is

Find the points on the curve $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ where the tangent line is vertical.

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2$ and $r = 3 - 2 \cos \theta,$ then A is

The parametric equations of the line segment from (1,0) to (0,1) with t $\in$ [0,1] are

If A is the area of the region bounded by one leaf of the lemniscates $r ^ { 2 } = 3 \sin 2 \theta,$ then A is

The length of the curve $x ( t ) = e ^ { t } \cos t , y ( t ) = e ^ { t } \sin t$ with $t \in [ 0 , \pi ]$ is

Let $x ( t ) = \ln t , y ( t ) = \frac { 1 } { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

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Exam 10: Parametric Equations; Polar Equations

The rectangular equation of the conic section r=35+4cos⁡θr = \frac { 3 } { 5 + 4 \cos \theta }r=5+4cosθ3​ is

The length of the curve x(t)=9−t3,y(t)=t2x ( t ) = 9 - t ^ { 3 } , y ( t ) = t ^ { 2 }x(t)=9−t3,y(t)=t2 with t∈[0,2]t \in [ 0,2 ]t∈[0,2] is

The length of r=6cos⁡θr = 6 \cos \thetar=6cosθ with x∈[0,π2]x \in \left[ 0 , \frac { \pi } { 2 } \right]x∈[0,2π​] is

The area of the surface generated by revolving the curve x(t)=3t,y=7t+1x ( t ) = 3 t , \quad y = \sqrt { 7 } t + 1x(t)=3t,y=7​t+1 with t∈[0,1]t \in [ 0,1 ]t∈[0,1] about the x-axis is

The conic section r=57+8sin⁡θr = \frac { 5 } { 7 + 8 \sin \theta }r=7+8sinθ5​ is

If A is the area of the intersection of the regions inside 6π−166 \pi - 166π−16 and outside r=3, r = \sqrt { 3 } \text {, }r=3​, then A is

If A is the area of the intersection of the regions enclosed by the graphs of r=2(1−cos⁡θ)r = 2 ( 1 - \cos \theta )r=2(1−cosθ) and r=2(1+cos⁡θ),r = 2 ( 1 + \cos \theta ),r=2(1+cosθ), then A is

The rectangular equation of the plane curve, with parametric equations x(t)=e−t,y(t)=3etx ( t ) = e ^ { - t } , y ( t ) = 3 e ^ { t }x(t)=e−t,y(t)=3et with t∈(−∞,∞)t \in ( - \infty , \infty )t∈(−∞,∞) , is

The polar equation that corresponds to the rectangular equation y=x2y = x ^ { 2 }y=x2 is

The rectangular equation that corresponds to the polar equation r2=θ\frac { r } { 2 } = \theta2r​=θ is

For a ≠\neq= 0, the polar curve r2=asin⁡2θr ^ { 2 } = a \sin 2 \thetar2=asin2θ is a

For a ≠\neq= 0, the polar curve r=a(1+2cos⁡θ)r = a ( 1 + 2 \cos \theta )r=a(1+2cosθ) is a

The area of the surface generated by revolving the curve x(t)=1−t2,y=3+2tx ( t ) = 1 - t ^ { 2 } , y = 3 + 2 tx(t)=1−t2,y=3+2t with t∈[0,1]t \in [ 0,1 ]t∈[0,1] about the x-axis is

If A is the area of the region bounded by one petal of the four-petal rose r=cos⁡2θ,r = \cos 2 \theta,r=cos2θ, then A is

Find the points on the curve x(t)=t31+t,y(t)=11+tx ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }x(t)=1+tt3​,y(t)=1+t1​ where the tangent line is vertical.

If A is the area of the intersection of the regions enclosed by the graphs of r=2r = 2r=2 and r=3−2cos⁡θ,r = 3 - 2 \cos \theta,r=3−2cosθ, then A is

The parametric equations of the line segment from (1,0) to (0,1) with t ∈\in∈ [0,1] are

If A is the area of the region bounded by one leaf of the lemniscates r2=3sin⁡2θ,r ^ { 2 } = 3 \sin 2 \theta,r2=3sin2θ, then A is

The length of the curve x(t)=etcos⁡t,y(t)=etsin⁡tx ( t ) = e ^ { t } \cos t , y ( t ) = e ^ { t } \sin tx(t)=etcost,y(t)=etsint with t∈[0,π]t \in [ 0 , \pi ]t∈[0,π] is

Let x(t)=ln⁡t,y(t)=1tx ( t ) = \ln t , y ( t ) = \frac { 1 } { t }x(t)=lnt,y(t)=t1​ be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

Preparing for Calculus

Limits and Continuity

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Applications of the Integral

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Multiple Integrals

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The rectangular equation of the conic section $r = \frac { 3 } { 5 + 4 \cos \theta }$ is

The length of the curve $x ( t ) = 9 - t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

The length of $r = 6 \cos \theta$ with $x \in \left[ 0 , \frac { \pi } { 2 } \right]$ is

The area of the surface generated by revolving the curve $x ( t ) = 3 t , \quad y = \sqrt { 7 } t + 1$ with $t \in [ 0,1 ]$ about the x-axis is

The conic section $r = \frac { 5 } { 7 + 8 \sin \theta }$ is

If A is the area of the intersection of the regions inside $6 \pi - 16$ and outside $r = \sqrt { 3 } \text {, }$ then A is

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2 ( 1 - \cos \theta )$ and $r = 2 ( 1 + \cos \theta ),$ then A is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } , y ( t ) = 3 e ^ { t }$ with $t \in ( - \infty , \infty )$ , is

The polar equation that corresponds to the rectangular equation $y = x ^ { 2 }$ is

The rectangular equation that corresponds to the polar equation $\frac { r } { 2 } = \theta$ is

For a $\neq$ 0, the polar curve $r ^ { 2 } = a \sin 2 \theta$ is a

For a $\neq$ 0, the polar curve $r = a ( 1 + 2 \cos \theta )$ is a

The area of the surface generated by revolving the curve $x ( t ) = 1 - t ^ { 2 } , y = 3 + 2 t$ with $t \in [ 0,1 ]$ about the x-axis is

If A is the area of the region bounded by one petal of the four-petal rose $r = \cos 2 \theta,$ then A is

Find the points on the curve $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ where the tangent line is vertical.

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2$ and $r = 3 - 2 \cos \theta,$ then A is

The parametric equations of the line segment from (1,0) to (0,1) with t $\in$ [0,1] are

If A is the area of the region bounded by one leaf of the lemniscates $r ^ { 2 } = 3 \sin 2 \theta,$ then A is

The length of the curve $x ( t ) = e ^ { t } \cos t , y ( t ) = e ^ { t } \sin t$ with $t \in [ 0 , \pi ]$ is

Let $x ( t ) = \ln t , y ( t ) = \frac { 1 } { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is