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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 #116 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-128,-78,-113).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(149.89,-148.64^{\circ},-113)$

$(149.89,31.36^{\circ},-113)$

$(206,-148.64^{\circ},-113)$

$(206,31.36^{\circ},-113)$

Correct answer:

$(149.89,-148.64^{\circ},-113)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-128,-78,-113)\\&r=\sqrt{(-128)^2+(-78)^2}=149.89\\&\theta=arctan(\frac{-78}{-128})=-148.64^{\circ}\\&z=-113\end{align*}$

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Example Question #117 : Cylindrical Coordinates

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-95,-78,-25).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(173,39.39^{\circ},-25)$

$(122.92,-140.61^{\circ},-25)$

$(122.92,39.39^{\circ},-25)$

$(173,-140.61^{\circ},-25)$

Correct answer:

$(122.92,-140.61^{\circ},-25)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-95,-78,-25)\\&r=\sqrt{(-95)^2+(-78)^2}=122.92\\&\theta=arctan(\frac{-78}{-95})=-140.61^{\circ}\\&z=-25\end{align*}$

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Example Question #205 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-136,121,134).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(182.04,41.66^{\circ},134)$

$(15,138.34^{\circ},134)$

$(15,41.66^{\circ},134)$

$(182.04,138.34^{\circ},134)$

Correct answer:

$(182.04,138.34^{\circ},134)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-136,121,134)\\&r=\sqrt{(-136)^2+(121)^2}=182.04\\&\theta=arctan(\frac{121}{-136})=138.34^{\circ}\\&z=134\end{align*}$

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Example Question #206 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-3,-3,-49).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(6,45^{\circ},-49)$

$(6,-135^{\circ},-49)$

$(4.24,-135^{\circ},-49)$

$(4.24,45^{\circ},-49)$

Correct answer:

$(4.24,-135^{\circ},-49)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-3,-3,-49)\\&r=\sqrt{(-3)^2+(-3)^2}=4.24\\&\theta=arctan(\frac{-3}{-3})=-135^{\circ}\\&z=-49\end{align*}$

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Example Question #207 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (120,-39,-117).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(126.18,-18^{\circ},-117)$

$(126.18,162^{\circ},-117)$

$(81,162^{\circ},-117)$

$(81,-18^{\circ},-117)$

Correct answer:

$(126.18,-18^{\circ},-117)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(120,-39,-117)\\&r=\sqrt{(120)^2+(-39)^2}=126.18\\&\theta=arctan(\frac{-39}{120})=-18^{\circ}\\&z=-117\end{align*}$

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Example Question #208 : 3 Dimensional Space

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (84,-33,-78).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(51,-21.45^{\circ},-78)$

$(90.25,158.55^{\circ},-78)$

$(51,158.55^{\circ},-78)$

$(90.25,-21.45^{\circ},-78)$

Correct answer:

$(90.25,-21.45^{\circ},-78)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(84,-33,-78)\\&r=\sqrt{(84)^2+(-33)^2}=90.25\\&\theta=arctan(\frac{-33}{84})=-21.45^{\circ}\\&z=-78\end{align*}$

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Example Question #1871 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-29,-121,-111).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(124.43,76.52^{\circ},-111)$

$(124.43,-103.48^{\circ},-111)$

$(150,76.52^{\circ},-111)$

$(150,-103.48^{\circ},-111)$

Correct answer:

$(124.43,-103.48^{\circ},-111)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-29,-121,-111)\\&r=\sqrt{(-29)^2+(-121)^2}=124.43\\&\theta=arctan(\frac{-121}{-29})=-103.48^{\circ}\\&z=-111\end{align*}$

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Example Question #1872 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (133,137,23).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(190.94,134.15^{\circ},23)$

$(270,134.15^{\circ},23)$

$(190.94,45.85^{\circ},23)$

$(270,45.85^{\circ},23)$

Correct answer:

$(190.94,45.85^{\circ},23)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(133,137,23)\\&r=\sqrt{(133)^2+(137)^2}=190.94\\&\theta=arctan(\frac{137}{133})=45.85^{\circ}\\&z=23\end{align*}$

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Example Question #1873 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-133,-80,-44).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(213,31.03^{\circ},-44)$

$(155.21,-148.97^{\circ},-44)$

$(155.21,31.03^{\circ},-44)$

$(213,-148.97^{\circ},-44)$

Correct answer:

$(155.21,-148.97^{\circ},-44)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-133,-80,-44)\\&r=\sqrt{(-133)^2+(-80)^2}=155.21\\&\theta=arctan(\frac{-80}{-133})=-148.97^{\circ}\\&z=-44\end{align*}$

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Example Question #1874 : Calculus 3

$\begin{align*}&\text{A point in space is located, in Cartesian coordinates, at } (-37,-93,-22).\\&\text{What is the position of this point in cylindrical coordinates?}\end{align*}$

Possible Answers:

$(100.09,68.3^{\circ},-22)$

$(130,-111.7^{\circ},-22)$

$(100.09,-111.7^{\circ},-22)$

$(130,68.3^{\circ},-22)$

Correct answer:

$(100.09,-111.7^{\circ},-22)$

Explanation:

$\begin{align*}&\text{When converting Cartesian coordinates of the form} (x,y,z)\\&\text{to cylindrical coordinates of the form }(r,\theta,z)\text{, the first and third terms}\\& \text{are the most straightforward: }\\& r=\sqrt{x^2+y^2}\\&z=z\\&\text{However, it is prudent to be carful when calculating }\theta.\\&\text{The formula for it is as follows:}\\& \theta=arctan(\frac{y}{x})\\&\text{Remember, it is important to be mindful of the signs of }x\text{ and }y\text{, bearing in mind}\\& \text{which quadrant the point lies in; this determines the value of }\theta.\\& \text{Negative }y\text{ values lead to a negative }\theta\text{; negative }x\\&\text{values lead to }|\theta|>90^{\circ}\\&\\&\text{For our coordinates: }(-37,-93,-22)\\&r=\sqrt{(-37)^2+(-93)^2}=100.09\\&\theta=arctan(\frac{-93}{-37})=-111.7^{\circ}\\&z=-22\end{align*}$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #116 : Cylindrical Coordinates

Example Question #117 : Cylindrical Coordinates

Example Question #205 : 3 Dimensional Space

Example Question #206 : 3 Dimensional Space

Example Question #207 : 3 Dimensional Space

Example Question #208 : 3 Dimensional Space

Example Question #1871 : Calculus 3

Example Question #1872 : Calculus 3

Example Question #1873 : Calculus 3

Example Question #1874 : Calculus 3

All Calculus 3 Resources

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