10.10.1 An Introduction to Arc Length

Question 1

An Introduction to Arc Length

Answer 1

• Arc length is the length of the curve.

* The arc length of a smooth curve given by the function f (x) between a and b is

Question 2

note

Answer 2

• When measuring how long a line is, you can just use a ruler or the distance formula. But curves are trickier. It would be good to have a way to measure their lengths. This length is called arc length.
• One way to think about arc length is to break a curve up into a lot of line segments. Then you can approximate the arc length by adding them all up.
• To find the exact arc length you have to use calculus.
• Pick two points on the curve that are very close to each other. The second point is a small change in x from the first point.
• The Pythagorean theorem tells you the length of the line segment connecting the two points.
• Notice that the length of the line segment is expressed in terms of the change in the two directions.
• To find the length of the entire curve, you must sum up the lengths of all the line segments.
• Factoring out a Δx moves the small change in x outside the radical sign.
• If you let Δx become arbitrarily small, then it acts like a dx. You can now find the arc length by integrating.
• Notice that the integral is different from the integral you
  would use to find the area under the curve.

Question 3

Set up the integral for the arc length of the curve y=x^3, where 1≤x≤2.

Answer 3

∫21√1+9x^4dx

Question 4

Given two smooth curves y1=f(x) andy2=−f(x), which of the following is the relation between the arc lengths L1 and L2 of the two curves on the interval[a, b]?

Answer 4

L1=L2

Question 5

Set up the integral for the arc length of the curve y=lnx, where 1≤x≤3

Answer 5

∫31⎷1+1/x^2dx

Question 6

Set up the integral for the arc length ofthe curve y=x^3+x, where 1≤x≤2.

Answer 6

∫21√1+(3x2+1)2 dx

Question 7

The proof of the formula for the length of a curve depends strongly on which of the following theorems?

Answer 7

Pythagorean theorem

Question 8

Given a smooth curve y = f (x) on the closed interval [a, b], which of the following formulas determines the arc length of the curve?

Answer 8

∫ba√1+[f′(x)]2dx

Question 9

Set up the integral for the arc length of the curve y=sinx, where 0≤x≤π.

Answer 9

∫π0√1+cos2xdx

Question 10

Set up the integral for the arc length of the curve y=√x, where 0≤x≤1.

Answer 10

∫1 0⎷1+1/4xdx

Question 11

Set up the integral for the arc length of the curve y=e2x, where 0≤x≤π

Answer 11

∫π0√1+4e^4xdx

Question 12

Set up the integral for the arc length ofthe curve y=x2, where 0≤x≤1.

Answer 12

∫10√1+(2x)2dx