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Calculus 3 : Cylindrical Coordinates

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Example Questions

Example Question #1 : Cylindrical Coordinates

Convert the following into Cylindrical coordinates.

$x=\sin(t)$

$y=\cos(t)$

$z=\tan(t)$

Possible Answers:

r=1

$\theta=\tan^{-1}(\cot(t))$

$z=\tan(t)$

r=1

$\theta=\tan^{-1}(\tan(t))$

$z=\tan(t)$

r=t

$\theta=\tan(\cot(t))$

$z=\tan(t)$

r=t

$\theta=\tan^{-1}(\tan(t))$

z=1

r=1

$\theta=\tan^{-1}(\cot(t))$

z=t

Correct answer:

r=1

$\theta=\tan^{-1}(\cot(t))$

$z=\tan(t)$

Explanation:

In order to convert to cylindrical coordinates, we need to recall the conversion equations.

$r=\sqrt{x^2+y^2}$

$\theta=\tan^{-1}(\frac{y}{x})$

z=z

Now lets apply this to our problem.

$r=\sqrt{(\sin(t))^2+(\cos(t))^2}=\sqrt{1}=1$

$\theta=\tan^{-1}\Big(\frac{\cos(t)}{\sin(t)}\Big)=\tan^{-1}(\cot(t))$

$z=\tan(t)$

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Example Question #2 : Cylindrical Coordinates

When converting rectangular coordinates to cylindrical coordinates, which variable remains fixed?

Possible Answers:

x,y

None of them are fixed.

Correct answer:

Explanation:

To convert a point (x,y,z) into cylindrical corrdinates, the transformation equations are

$x = r\cos\theta$

$y = r\sin\theta$

z = z .

Choices for $r, \theta$ may vary depending on the situation, but the coordinate remains the same.

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Example Question #2 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (5,12,7) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(17,67.38^{\circ},7)$

$(13,67.38^{\circ},7)$

$(13,112.62^{\circ},7)$

$(17,112.62^{\circ},7)$

$(17,67.38^{\circ},13)$

Correct answer:

$(13,67.38^{\circ},7)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative values by convention create a negative $\theta$ , while negative values lead to $|\theta|>90^{\circ};\frac{\pi}{2}$

For our coordinates (5,12,7)

$r=\sqrt{x^2+y^2}\rightarrow13$

$\theta=arctan(\frac{y}{x})\rightarrow67.38^{\circ}$ (Bearing in mind sign convention)

z=7

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Example Question #2 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (4,-4,8) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(5.66,-135^{\circ},8)$

$(5.66,135^{\circ},8)$

$(5.66,-45^{\circ},8)$

$(5.66,45^{\circ},8)$

Correct answer:

$(5.66,-45^{\circ},8)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (4,-4,8)

$r=\sqrt{x^2+y^2}\rightarrow5.66$

$\theta=arctan(\frac{y}{x})\rightarrow -45^{\circ}$ (Bearing in mind sign convention)

z=8

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Example Question #2 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at $(1,\sqrt{3},4)$ . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(3,120^{\circ},4)$

$(3,60^{\circ},4)$

$(4,-120^{\circ},2)$

$(2,120^{\circ},4)$

$(2,60^{\circ},4)$

Correct answer:

$(2,60^{\circ},4)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates $(1,\sqrt{3},4)$

$r=\sqrt{x^2+y^2}\rightarrow 2$

$\theta=arctan(\frac{y}{x})\rightarrow 60^{\circ}$ (Bearing in mind sign convention)

z=4

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Example Question #3 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-9,40,11) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(41,-77.32^{\circ},11)$

$(41,77.32^{\circ},11)$

$(41,102.68^{\circ},11)$

$(41,105.61^{\circ},11)$

$(41,74.39^{\circ},11)$

Correct answer:

$(41,102.68^{\circ},11)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-9,40,11)

$r=\sqrt{x^2+y^2}\rightarrow41$

$\theta=arctan(\frac{y}{x})\rightarrow 102.68^{\circ}$ (Bearing in mind sign convention)

z=11

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Example Question #2 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-3,-4,10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(5,53.13^{\circ},10)$

$(5,126.87^{\circ},10)$

$(10,126.87^{\circ},5)$

$(5,-126.87^{\circ},10)$

$(10,53.13^{\circ},5)$

Correct answer:

$(5,-126.87^{\circ},10)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-3,-4,10)

$r=\sqrt{x^2+y^2}\rightarrow5$

$\theta=arctan(\frac{y}{x})\rightarrow -126.87^{\circ}$ (Bearing in mind sign convention)

z=10

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Example Question #8 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-4,8,10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.94,-41.62^{\circ},10)$

$(-8.94,-63.44^{\circ},10)$

$(8.94,-63.44^{\circ},10)$

$(8.94,41.62^{\circ},10)$

$(8.94,116.57^{\circ},10)$

Correct answer:

$(8.94,116.57^{\circ},10)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-4,8,10)

$r=\sqrt{x^2+y^2}\rightarrow8.94$

$\theta=arctan(\frac{y}{x})\rightarrow 116.57^{\circ}$ (Bearing in mind sign convention)

z=10

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Example Question #9 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (6,8,10) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(10,-126.87^{\circ},10)$

$(10,53.13^{\circ},10)$

$(10,53.13^{\circ},20)$

$(10,-71.19^{\circ},10)$

$(10,108.81^{\circ},10)$

Correct answer:

$(10,53.13^{\circ},10)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (6,8,10)

$r=\sqrt{x^2+y^2}\rightarrow10$

$\theta=arctan(\frac{y}{x})\rightarrow 53.13^{\circ}$ (Bearing in mind sign convention)

z=10

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Example Question #10 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-5,-5,-5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(7.07,-135^{\circ},-5)$

$(7.07,70^{\circ},-5)$

$(7.07,45^{\circ},-5)$

$(7.07,-120^{\circ},-5)$

$(7.07,110^{\circ},-5)$

Correct answer:

$(7.07,-135^{\circ},-5)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-5,-5,-5)

$r=\sqrt{x^2+y^2}\rightarrow7.07$

$\theta=arctan(\frac{y}{x})\rightarrow -135^{\circ}$ (Bearing in mind sign convention)

z=-5

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Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Cylindrical Coordinates

Example Question #2 : Cylindrical Coordinates

Example Question #2 : Cylindrical Coordinates

Example Question #2 : Cylindrical Coordinates

Example Question #2 : Cylindrical Coordinates

Example Question #3 : Cylindrical Coordinates

Example Question #2 : Cylindrical Coordinates

Example Question #8 : Cylindrical Coordinates

Example Question #9 : Cylindrical Coordinates

Example Question #10 : Cylindrical Coordinates

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