Calculus II MAT 146 Integration Applications: Arc Length

[Pages:3]Calculus II MAT 146 Integration Applications: Arc Length Again we use a definite integral to sum an infinite number of measures, each infinitesimally small. We seek to determine the length of a curve that represents

the graph of some real-valued function f, measuring from the point (a, f (a)) on the curve to the point (b, f (b)) on the curve. ( ( )) The graph of y = f is shown. We seek to determine the length of the curve, known as arc length, from the point a, f a ( ( )) on the curve to the point b, f b .

( ) ( ) ( ) Together with endpoints a, f (a) ( ) and b, f b , we select additional points ( ) xi, f xi on the curve. Here, we've

created n line segments. The length of the ith line segment is denoted by Li, with 1 i n.

Concentrate on the segment whose length is Li. Determine the slope of that segment:

slope of segment whose length is Li = mi

= f (xi ) ! f (xi!1)

xi ! xi!1

= "yi "xi

#

mi

=

"yi "xi

# mi $ "xi = "yi

We know by the Mean Value Theorem that there exists an xi*, with xi?1 xi* xi-1, such

that mi = f !(xi*) . We use this fact, and the distance formula, to calculate the typical

segment length Li:

( ) Li = (xi ! ) xi!1 2 + f (xi ) ! f (xi!1) 2

= ("xi )2 + ("yi )2

= ("xi )2 + ("yi )2

= ("xi )2 + (mi"xi )2

= ("xi )2 + ( f #(xi*)"xi )2

=

("xi

)2

$%&1

+

(

f

#(

xi*

))2

' ()

= "xi 1+ ( f #(xi*))2

We now let the number of segments between (a, f(a)) and (b, f(b)) grow without bound.

That is, we let n--> . We can now write a definite integral to add all the lengths Li:

total length of the curve from x = a to x = b :

b

&

1+"# f !(x)$%2 dx

a

Example #1: Determine the length of the curve y = x2 from x = 0 to x = 3.

b

We use the arc length formula &

1+"# f !(x)$%2 dx . Here, a = 0 and b = 3. The function

a

y = f (x) = x2 has derivative f !(x) = 2x , so

b

arc length = &

1+"# f !(x)$%2 dx

a

3

=&

1+ [2x]2 dx

0

3

= & 1+ 4x2 dx

0

( ) = 1 ln 37 + 6 + 3 37

4

2

' 9.747089

Example #2: What is the arc length for y = x3 + 1 , 1 ! x !1 ?

6 2x 2

b

Again, use the arc length formula &

1+"# f !(x)$%2 dx . Here, a = ? and b = 1, with

a

dy dx

=

x2 2

!

1 2x2

.

b

arc length = &

1+"# f !(x)$%2 dx

a

1

=&

1 2

" 1+(

#

x2 2

'

1 2x2

$2 ) %

dx

= 31 48

Example #3: For x = ln(1! y2 ) , determine the length of arc x(y) for 0 y ?.

d

Here, we use the arc length formula with y as the independent variable: &

1+"#h!(y)$%2 dy .

c

Here, we have c = 0 and d = ?, with

dx dy

=

"1 #$1! y2

%

' (!2 y)

&

=

!2 y 1! y2

.

d

arc length = &

1+"# f !(y)$%2 dy

c

1 2

=&

0

1

+

" '2y #(1' y2

$2 ) %

dy

* 0.5986123

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download