section 3 arc length. definition let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t...

Section 3 Arc Length

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Section 3

Arc Length

Definition

Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t1,t2]

The arc length of the curve having the above parametric equations on the interval [t1,t2] is defined to

be:

dtt

tdtdy

dtdx

22 )()(

Page 3: Section 3 Arc Length. Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above

Example (1)

Find the arc length L of the curve:

x=rcost, y=rsint ; 0 ≤ t ≤ 2π, where r is a constant ( what does that represent?)

Solution:

2)cos()sin[(2

Why

rdttrtrL

Page 4: Section 3 Arc Length. Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above

Example (2)

Find the arc length L of the curve:

x=r(t-sint), y=r(1-cost) ; 0 ≤ t ≤ 2π, where r is a constant

Solution:

dx/dt = ?

dy/dt=?

rrrt

dtt

Whydtt

Whydttr

WhydttrtrL

8]1)1([4)0coscos[4)2

cos(22

2sin22

?**2

sin2

?*2

sin22

?cos12

?cos22

?)sin(])cos1([

222

Page 6: Section 3 Arc Length. Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above

20;2

sin2

sin

?20;02

sin

2020

?**2

sin22

cos12

sin2

2sin21

2sin)

2sin1(

2sin

2coscos:

?*2

sin22cos12

22222

ttt

Whytt

Answer

Whydtt

rdtt

ttttttAnswer

Whydtt

rdttr

DefinitionLet x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t1,t2]

Assume that g is non-negative and the derivatives of f and g are continuos on that interval.

The surface area resulting from the rotation the curve having the above parametric equations about the x-axis on the interval [t1,t2] is defined to be:

dtds

wheredsydtyS

dtdy

dtdx

tdtdy

dtdx

)()(

,,2)()(22

Page 8: Section 3 Arc Length. Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above

Example (1)

Find the surface area of the sphere with radius r !

Solution:

Consider the semicircle:

x = r cost , y = r sint ; 0 ≤ t ≤ π, where r is a

constant

The rotation of this semicircle results in a sphere

of radius r

Page 9: Section 3 Arc Length. Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above

220

4)11(2cos2

?sin2

?)cos()sin(sin2

)()(2

cos,sin

0;sin,cos

rrtr

Whydttr

Whydttrtrtr

dtyS

trdt

dytr

ttrytrx

tdtdy

dtdx

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