Arc Length and Speed What is the length of this curve? a polygonal path inscribed in the curve C Figure 10.7.1 Po Figure 10.7.2 A curve C parametrized by a pair of functions x=x(),y=(),t∈[a,]. We want to determine the length of C. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed A curve C parametrized by a pair of functions x = x(t), y = y(t), t [a, b]. We want to determine the length of C
Arc Length and Speed The length of the path C traced out by a pair of continuously differentiable functions x=x(1), y=0t∈[a,b] is given by the formula (10.7.1) L(C)= /[x'()+[y'()P dt. Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed The length of the path C traced out by a pair of continuously differentiable functions x = x(t), y = y(t) t [a, b] is given by the formula
Arc Length and Speed Suppose that C is the graph of y=f(x).x [a,b]. C can be parametrized by x()=t,)=f(0t∈[a,b] Since x(t)=1 and y'(t)=f'(t),we have L(C)=S+[r(Jd Replacing t by x,we can write: (10.7.2) The length of the graph off= √1+[f"xPd Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed Suppose that C is the graph of y = f (x), x [a, b]. C can be parametrized by x(t) = t, y(t) = f (t) t [a, b]. Since x’(t) = 1 and y’ (t) = f ’ (t) , we have ( ) ( ) 2 1 b a L C f t dt = + Replacing t by x, we can write:
Arc Length and Speed Example 1 f(x)=x+x Find the length of the graph from x=1 to x=3 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed Example 1 Find the length of the graph from x = 1 to x = 3
Arc Length and Speed Example 2 The graph of the function f(x)=x2 from x=0 to x=1 is parabolic arc.Find the length of this arc. Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed Example 2 The graph of the function f (x) = x 2 from x = 0 to x = 1 is parabolic arc. Find the length of this arc
Arc Length and Speed Example 3 For fixed a>0,the equation r=a represents a circle of radius a.The circle is traced out once as 0 ranges from 0 to 2n.Find the length of the curve(the circumference of the circle). Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed Example 3 For fixed a > 0, the equation r = a represents a circle of radius a. The circle is traced out once as θ ranges from 0 to 2π. Find the length of the curve (the circumference of the circle)
Arc Length and Speed Example 4 Calculate the arclength of the cardioid r=a(1-cos),where a>0, 0 lies on the interval [0.2xl. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed Example 4 Calculate the arclength of the cardioid r = a(1-cosθ), where a > 0, θ lies on the interval [0, 2π]
Arc Length and Speed The Geometric Significance of dx/ds and dy/ds tangent b Figure 10.7.3 Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed The Geometric Significance of dx/ds and dy/ds
d (107.40 多 =c0s必and sinax where is the incination of the ds ds tangent at the pont(). Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Arc Length and Speed Example 5 The position of a particle at time t is given by the parametric equations x(t)=3cos2t,y(t)=4 sin 2t tE[0,2n]. Find the speed of the particle at time t and determine the times when the speed is a maximum and when it is a minimum. 3,01 Figure 10.7.4 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Arc Length and Speed Example 5 The position of a particle at time t is given by the parametric equations. Find the speed of the particle at time t and determine the times when the speed is a maximum and when it is a minimum