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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #72 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (3,24,5) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(24.19,82.87^{\circ},5)$

$(27,-82.87^{\circ},5)$

$(27,97.13^{\circ},5)$

$(27,82.87^{\circ},5)$

$(24.19,97.13^{\circ},5)$

Correct answer:

$(24.19,82.87^{\circ},5)$

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

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 (3,24,5)

$\begin{align*}&r=\sqrt{(3)^2+(24)^2}=24.19\\&\theta=arctan(\frac{24}{3})=82.87^{\circ}\\&z=5\end{align*}$

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Example Question #73 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-2,-20,-22) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(20.1,-95.71^{\circ},-22)$

$(22,84.29^{\circ},-22)$

$(20.1,-84.29^{\circ},-22)$

$(20.1,95.71^{\circ},-22)$

$(20.1,84.29^{\circ},-22)$

Correct answer:

$(20.1,-95.71^{\circ},-22)$

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 (-2,-20,-22)

$\begin{align*}&r=\sqrt{(-2)^2+(-20)^2}=20.1\\&\theta=arctan(\frac{-20}{-2})=-95.71^{\circ}\\&z=-22\end{align*}$

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Example Question #81 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (32,-11,37) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(21,-18.97^{\circ},37)$

$(33.84,-18.97^{\circ},37)$

$(33.84,161.03^{\circ},37)$

$(21,161.03^{\circ},37)$

Correct answer:

$(33.84,-18.97^{\circ},37)$

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 (32,-11,37)

$\begin{align*}&r=\sqrt{(32)^2+(-11)^2}=33.84\\&\theta=arctan(\frac{-11}{32})=-18.97^{\circ}\\&z=37\end{align*}$

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Example Question #82 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-2,-16,-15) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(18,82.87^{\circ},-15)$

$(16.12,82.87^{\circ},-15)$

$(16.12,-97.13^{\circ},-15)$

$(18,-97.13^{\circ},-15)$

Correct answer:

$(16.12,-97.13^{\circ},-15)$

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 (-2,-16,-15)

$\begin{align*}&r=\sqrt{(-2)^2+(-16)^2}=16.12\\&\theta=arctan(\frac{-16}{-2})=-97.13^{\circ}\\&z=-15\end{align*}$

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Example Question #83 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-26,0,45) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(26,180^{\circ},45)$

$(26,45^{\circ},45)$

$(26,-45^{\circ},45)$

$(26,0^{\circ},45)$

Correct answer:

$(26,180^{\circ},45)$

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 (-26,0,45)

$\begin{align*}&r=\sqrt{(-26)^2+(0)^2}=26\\&\theta=arctan(\frac{0}{-26})=180^{\circ}\\&z=45\end{align*}$

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Example Question #84 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-8,-3,-23) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(8.54,-159.44^{\circ},-23)$

$(11,20.56^{\circ},-23)$

$(11,-159.44^{\circ},-23)$

$(8.54,20.56^{\circ},-23)$

Correct answer:

$(8.54,-159.44^{\circ},-23)$

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 (-8,-3,-23)

$\begin{align*}&r=\sqrt{(-8)^2+(-3)^2}=8.54\\&\theta=arctan(\frac{-3}{-8})=-159.44^{\circ}\\&z=-23\end{align*}$

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Example Question #171 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (45,30,-29) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(54.08,33.69^{\circ},-29)$

$(54.08,146.31^{\circ},-29)$

$(75,33.69^{\circ},-29)$

$(75,146.31^{\circ},-29)$

Correct answer:

$(54.08,33.69^{\circ},-29)$

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 (45,30,-29)

$\begin{align*}&r=\sqrt{(45)^2+(30)^2}=54.08\\&\theta=arctan(\frac{30}{45})=33.69^{\circ}\\&z=-29\end{align*}$

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Example Question #172 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (48,17,29) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(65,160.5^{\circ},29)$

$(50.92,160.5^{\circ},29)$

$(50.92,19.5^{\circ},29)$

Correct answer:

$(50.92,19.5^{\circ},29)$

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 (48,17,29)

$\begin{align*}&r=\sqrt{(48)^2+(17)^2}=50.92\\&\theta=arctan(\frac{17}{48})=19.5^{\circ}\\&z=29\end{align*}$

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Example Question #173 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-12,7,-14) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(5,149.74^{\circ},-14)$

$(13.89,149.74^{\circ},-14)$

$(13.89,-30.26^{\circ},-14)$

$(5,-30.26^{\circ},-14)$

Correct answer:

$(13.89,149.74^{\circ},-14)$

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 (-12,7,-14)

$\begin{align*}&r=\sqrt{(-12)^2+(7)^2}=13.89\\&\theta=arctan(\frac{7}{-12})=149.74^{\circ}\\&z=-14\end{align*}$

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Example Question #1831 : Calculus 3

A point in space is located, in Cartesian coordinates, at (17,21,15) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(38,51.01^{\circ},15)$

$(27.02,51.01^{\circ},15)$

$(27.02,128.99^{\circ},15)$

$(38,128.99^{\circ},15)$

Correct answer:

$(27.02,51.01^{\circ},15)$

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 (17,21,15)

$\begin{align*}&r=\sqrt{(17)^2+(21)^2}=27.02\\&\theta=arctan(\frac{21}{17})=51.01^{\circ}\\&z=15\end{align*}$

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #72 : Cylindrical Coordinates

Example Question #73 : Cylindrical Coordinates

Example Question #81 : Cylindrical Coordinates

Example Question #82 : Cylindrical Coordinates

Example Question #83 : Cylindrical Coordinates

Example Question #84 : Cylindrical Coordinates

Example Question #171 : 3 Dimensional Space

Example Question #172 : 3 Dimensional Space

Example Question #173 : 3 Dimensional Space

Example Question #1831 : Calculus 3

All Calculus 3 Resources

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