# Calculus 3 : Arc Length and Curvature

## Example Questions

### Example Question #1 : Arc Length And Curvature

Determine the length of the curve , on the interval

Explanation:

First we need to find the tangent vector, and find its magnitude.

Now we can set up our arc length integral

### Example Question #2 : Arc Length And Curvature

Determine the length of the curve , on the interval

Explanation:

First we need to find the tangent vector, and find its magnitude.

Now we can set up our arc length integral

### Example Question #3 : Arc Length And Curvature

Find the length of the curve , from , to

Explanation:

The formula for the length of a parametric curve in 3-dimensional space is

Taking dervatives and substituting, we have

. Factor a  out of the square root.

. "Uncancel" an  next to the . Now there is a perfect square inside the square root.

. Factor

. Take the square root, and integrate.

### Example Question #4 : Arc Length And Curvature

Find the length of the arc drawn out by the vector function  with  from to .

Explanation:

To find the arc length of a function, we use the formula

.

Using we have

### Example Question #5 : Arc Length And Curvature

Evaluate the curvature of the function  at the point .

Explanation:

The formula for curvature of a Cartesian equation is . (It's not the easiest to remember, but it's the most convenient form for Cartesian equations.)

We have , hence

and .

### Example Question #6 : Arc Length And Curvature

Find the length of the parametric curve

for .

Explanation:

To find the solution, we need to evaluate

.

First, we find

.

So we have a final expression to integrate for our answer

### Example Question #7 : Arc Length And Curvature

Determine the length of the curve given below on the interval 0<t<2

Explanation:

The length of a curve r is given by:

To solve:

### Example Question #8 : Arc Length And Curvature

Find the arc length of the curve

on the interval

Explanation:

To find the arc length of the curve function

on the interval

For the curve function in this problem we have

and following the arc length formula we solve for the integral

Hence the arc length is

### Example Question #601 : Calculus 3

Find the arc length of the curve function

On the interval

Round to the nearest tenth.

Explanation:

To find the arc length of the curve function

on the interval

For the curve function in this problem we have

and following the arc length formula we solve for the integral

Using u-substitution, we have

and

The integral then becomes

Hence the arc length is

### Example Question #1 : Arc Length And Curvature

Given that a curve is defined by , find the arc length in the interval