Lesson 14 - Length of an Arc

Lesson 14 - Length of an Arc - = = à à à à ∠LENGTH OF AN...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
TOPIC LENGTH OF AN ARC
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
OBJECTIVES discuss the formula for computing length of an arc; compute the length of an arc.
Background image of page 2
Length of curves To find the length of the arc of the curve y = f ( x ) between x = a and x = b let δ s be the length of a small element of arc so that: 2 2 2 2 ( ) ( ) ( ) so 1 s x y s y x x 2245 +   2245 +  ÷  
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Length of curves In the limit as the arc length δ s approaches zero: and so: 2 1 ds dy dx dx   = +  ÷   2 1 b x a b x a ds s dx dx dy dx dx = = =   = +  ÷  
Background image of page 4
Length of curves – parametric equations Instead of changing the variable of the integral as before when the curve is defined in terms of parametric equations, a special form of the result can be established which saves a deal of working when it is used. Let: 2 1 2 2 2 2 2 2 2 2 2 2 ( ) and ( ). As before ( ) ( ) ( ) so so as 0 and t t t y f t x F t s x y s x y t t t t ds dx dy dx dy s dt dt dt dt dt dt δ = = = 2245 +       2245 + →  ÷  ÷  ÷              
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = + = + ÷ ÷ ÷ ÷ ∫ LENGTH OF AN ARC A. Rectangular Coordinates f(x) y , 1 2 = + = dx dx dy ds g(y) x , dy dy dx 1 ds 2 = + = B. Parametric Form dt dt dy dt dx ds 2 2 + = when x=x(t), y=y(t); where t is a parameter 1. 2. C. Polar Coordinates f(r) , dr dr d r 1 ds 2 2 = Ξ Ξ + = ) g( r , d d dr r ds 2 2 Ξ = Ξ Ξ + = 2. 1. EXAMPLE Find the length of the arc of each of the following: 2 3 3 3 t y t t x =-= 1. from t = 0 to t = 1 2. t e y t e x t t sin cos = = from t = 0 to t = 4 4. Length of the arc of the semicircle 2 2 2 a y x = + 3. HW #3 Find the exact arc length of the curve over the interval....
View Full Document

This note was uploaded on 10/19/2011 for the course MATH 22 taught by Professor Ma'amzapanta during the Fall '11 term at MapĂșa Institute of Technology.

Page1 / 8

Lesson 14 - Length of an Arc - = = à à à à ∠LENGTH OF AN...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online