# Calculus 3 : Parametric Curves

## Example Questions

### Example Question #1 : Parametric Curves

Find the length of the parametric curve described by

from  to .

Explanation:

There are several ways to solve this problem, but the most effective would be to notice that we can derive the following-

Hence

Therefore our curve is a circle of radius , and it's circumfrence is . But we are only interested in half that circumfrence ( is from  to , not .), so our answer is .

Alternatively, we could've found the length using the formula

.

### Example Question #2 : Parametric Curves

Find the coordinates of the curve function

when .

Explanation:

To find the coordinates, we set  into the curve function.

We get

and thus

### Example Question #2 : Parametric Curves

Find the coordinates of the curve function

when

Explanation:

To find the coordinates, we evaluate the curve function for

As such,

### Example Question #3 : Parametric Curves

Find the coordinates of the curve function

when

Explanation:

To find the coordinates, we evaluate the curve function for

As such,

### Example Question #4 : Parametric Curves

Find the equation of the line passing through the two points, given in parametric form:

Explanation:

To find the equation of the line passing through these two points, we must first find the vector between them:

This was done by finding the difference between the x, y, and z components for the vectors. (This can be done in either order, it doesn't matter.)

Now, pick a point to be used in the equation of the line, as the initial point. We write the equation of line as follows:

The choice of initial point is arbitrary.

### Example Question #6 : Parametric Curves

Find the coordinate of the parametric curve when

Explanation:

To find the coordinates of the parametric curve we plug in for

.

As such the coordinates are