# Calculus 2 : Graphing Vectors

## Example Questions

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### Example Question #1 : Graphing Vectors

What is the arclength, from  to , of the curve:

Hint:

Explanation:

Arclength is given by the formula:

We should find dy/dx first, which we find to be:

Now let's proceed with the integral:

(here we apply the integration described in the hint)

which is obtained by evaluating at both boundaries.

### Example Question #1 : Graphing Vectors

Find the area of the polar equation:

Explanation:

When you plot the graph of   , the bounds are between   and .

Use the area formula  for polar equations:

And so we find the area to be:

### Example Question #1 : Graphing Vectors

The graph of the vector valued function

looks most like which of the following scalar functions graphed on a Cartesian coordinate system?

Explanation:

To determine this, we must convert our parametric form back into Cartesian form

To do this, we will solve for  and reverse substitute.

Since we know the parameterization for y, we substitute  in for

### Example Question #1 : Graphing Vectors

The graph of the vector function  can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #2 : Graphing Vectors

The graph of the vector function  can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #1 : Graphing Vectors

The graph of the vector function  can also be represented by the graph of which of the following functions in rectangular form?

None of the above

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #1 : Graphing Vectors

The graph of the vector function   can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #8 : Graphing Vectors

The graph of the vector function   can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #2 : Graphing Vectors

The graph of the vector function   can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #1 : Graphing Vectors

The graph of the vector function can also be represented by the graph of which of the following functions in rectangular form?