9 FURTHER APPLICATIONS OF INTEGRATION

SECTION 9.1
ARC LENGTH
6.
y
y
2.1
2
2
1.9
0
¹ 2¹
x
1.8
3 3.1 3.2 3.3 x
Notice that as x approaches π, the curve wiggles faster and faster, with decreasing amplitude. The limit as
x → π does exist, and is 2. We can show that L =∞either geometrically, or by comparing the resultant
∫ π
dx
integral to the improper integral 2
0 x − π .
A = 2 ∫ [ ( ) ]
π
1
0
(x − π) sin + 2 dx ≈ 17.385. (Note that this is an improper integral.)
x − π
Note that we can change the definition of f slightly, to “flip” the left half of the curve:
⎧
( ) 1
⎪⎨ − (x − π) sin + 2 if x < π
x − π
f (x) =
( ) 1
⎪⎩ (x − π) sin + 2 if x > π
x − π
y
2
0
¹ 2¹ x
The area of this new island (“Hpela”) is now the familiar 4π, and the perimeter is still infinite.
GROUP WORK 2: Cable Guy
This exercise requires the students to make several assumptions. For example, when computing the length of
cable needed for the sample segment shown on the left, they will have to decide which of the three figures on
the right most accurately represents the cross-sectional side view of the lake bottom:
10
20
10
20
10
20
10
20
Have them make an assumption and provide justification for it. Also remind students that the path of steepest
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