# Calculus 3 : Cylindrical Coordinates

## Example Questions

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### Example Question #1 : Cylindrical Coordinates

Convert the following into Cylindrical coordinates.

Explanation:

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

Now lets apply this to our problem.

### Example Question #2 : Cylindrical Coordinates

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

None of them are fixed.

Explanation:

To convert a point  into cylindrical corrdinates, the transformation equations are

.

Choices for  may vary depending on the situation, but the  coordinate remains the same.

### Example Question #3 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #4 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #5 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #6 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #7 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #8 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #9 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

### Example Question #10 : Cylindrical Coordinates

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

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

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

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 :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to

For our coordinates

(Bearing in mind sign convention)

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