Flashcards in 10.10.1 An Introduction to Arc Length Deck (12):
An Introduction to Arc Length
• 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
- 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.
Set up the integral for the arc length of the curve y=x^3, where 1≤x≤2.
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]?
Set up the integral for the arc length of the curve y=lnx, where 1≤x≤3
Set up the integral for the arc length ofthe curve y=x^3+x, where 1≤x≤2.
The proof of the formula for the length of a curve depends strongly on which of the following theorems?
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?
Set up the integral for the arc length of the curve y=sinx, where 0≤x≤π.
Set up the integral for the arc length of the curve y=√x, where 0≤x≤1.
Set up the integral for the arc length of the curve y=e2x, where 0≤x≤π