# Calculus 3 : Arc Length and Curvature

## Example Questions

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### Example Question #11 : Arc Length And Curvature

Find the arc length of the parametric curve

on the interval .

Round to the nearest tenth.

Explanation:

To find the arc length of the curve function

on the interval

we follow the formula

For the curve function in this problem we have

and following the arc length formula we solve for the integral

And using u-substitution, we set  and then solve the integral

Which is approximately

units

### Example Question #12 : Arc Length And Curvature

Determine the curvature of the vector .

Explanation:

Using the formula for curvature ,  , and . Plugging into the formula, we get

### Example Question #13 : Arc Length And Curvature

Find the arc length of the given curve on the interval :

Explanation:

The arc length on the interval  is given by

, where  is the magnitude of the tangent vector.

The tangent vector is given by

The magnitude of the vector is

This is the integrand.

Finally, integrate:

### Example Question #14 : Arc Length And Curvature

Determine the arc length of the following vector on the interval :

Explanation:

The arc length of a curve on some interval  is given by

where  is the tangent vector to the curve.

The tangent vector to the curve is found by taking the derivative of each component:

The magnitude of the vector is found by taking the square root of the sum of the squares of each component:

Now, plug this into the integral and integrate:

### Example Question #15 : Arc Length And Curvature

Given that

Find an expression for the curvature of the given conic

Explanation:

Step 1: Find the first and the second derivative

Step 2:

Radius of curvature is given by

Now substitute the calculated expressions into the equation to find the final answer

### Example Question #16 : Arc Length And Curvature

Find an integral for the arc length of

on the interval  (Set up, DO NOT SOLVE)

Explanation:

Step 1:

Find the first derivative of the function

Step 2:

Use the formula to calculate arc length

### Example Question #17 : 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 #18 : 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

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