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 #1 : 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 #2 : 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 #3 : Arc Length And Curvature

Find the length of the curve , from , to 

Possible Answers:

None of the other answers

Correct answer:

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 .

Possible Answers:

None of the other answers

Correct answer:

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 .

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

Explanation:

To find the solution, we need to evaluate

.

First, we find 

, which leads to 

.

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

Possible Answers:

Correct answer:

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 

 

 

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

Hence the arc length is 

Example Question #9 : Arc Length And Curvature

Find the arc length of the curve function

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

Using u-substitution, we have

and

The integral then becomes

Hence the arc length is 

Example Question #10 : Arc Length And Curvature

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

Possible Answers:

Correct answer:

Explanation:

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