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AP Calculus AB: 10.10.1 An Introduction to Arc Length

Mathematics12 CardsCreated 8 months ago

This content introduces the concept of arc length as the exact length of a curve, derived using calculus. It explains how arc length is calculated by integrating the square root of 1 plus the square of the derivative of a function, and includes examples of setting up arc length integrals for various curves.

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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

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Key Terms

Term
Definition

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 ...

note

  • 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 wa...

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

∫21√1+9x^4dx

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]?

L1=L2

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

∫31⎷1+1/x^2dx

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

∫21√1+(3x2+1)2 dx

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AP Calculus AB: 10.10.1 An Introduction to Arc Length
TermDefinition

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

note

  • 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.

∫21√1+9x^4dx

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]?

L1=L2

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

∫31⎷1+1/x^2dx

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

∫21√1+(3x2+1)2 dx

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

Pythagorean theorem

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?

∫ba√1+[f′(x)]2dx

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

∫π0√1+cos2xdx

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

∫1 0⎷1+1/4xdx

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

∫π0√1+4e^4xdx

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

∫10√1+(2x)2dx