Section 6.4: Arc Length
Elementary geometry doesn’t help to find the
length of a curve that wiggles like this.
So, let’s break it into pieces we can work with.
Arc Length:
y
s 2  x 2  y 2
x
s  x  y
2
Idea :
 y  2
 1  2  x
 x 
2
2
y
dy

x
dx
y
 1 


x


2
dy
Reason MVT 
 f '( xk* )
dx
b

a
1   f '  x   dx
2
x
f (x) = sinx, 0  x  


0

1   cos x  dx
2
2
f (x) = x on [0,3]
3

1   2x  dx
2
0
Or, f (y) =
y on [0,9]
9

0
 1
1 
2 y

2

 dy

Rotate the graph of x 2 about the x-axis. Find surface area.
9
3
 2 ( x )
2
0
3
1   2 x  dx
2
9
Rotate about y =  4
x2
3
-4
3
 2 ( x
0
2
 4) 1   2 x  dx
2
15
Rotate about y = 15
9
x2
3
3
 2 (15  x )
2
0
1   2 x  dx
2