Calculus 3 : Arc Length and Curvature

Study concepts, example questions & explanations for Calculus 3

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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.

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 :

Possible Answers:

Correct answer:

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 :

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

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)

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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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