Fall 2010 Math 152
2
Week in Review VI
Section 9.3
Key Points:
courtesy: David J. Manuel
1. Basic Formula: ds2 = dx2 + dy 2
s
2
ˆ b
dy
(a) f (x) : s =
dx
1+
dx
a
s
ˆ d 2
dx
(b) g(y) s =
+ 1 dy
dy
c
(covering 8.9, 9.3)
1
Section 8.9
(c) parametrized:
s
s
ˆ b 2 2
dy
dx
+
dt
dt
dt
a
Key Points:
1. Improper Integrals: unbounded regions
(a) “Type
I”
(unbounded
x):
ˆ ∞
ˆ t
f (x) dx = lim
f (x) dx (likewise
t→∞
a
Examples:
a
1. Find the length of the curve y = ln(sec x)
π
from x = 0 to x = .
4
√
2. Find the length of the curve y = 2 x from
x = 0 to x = 1.
with −∞)
(b) “Type II” (unbounded y): if f unbounded at x = a,
ˆ b
ˆ b
f (x) dx = lim+
f (x) dx (likewise
t→a
a
t
with b− )
3. Find the length of the curve y =
from x = 0 to x = ln 2
Examples:
5. Find the length of the curve parametrized by
x = a cos3 t, y = a sin3 t from t = 0 to t = 2π.
0
∞
(b)
2
(c)
ˆ
1
√ dx
x
5
(d)
ˆ
∞
(e)
ˆ
1
dx
(x − 2)2
∞
(f)
ˆ
∞
(g)
ˆ
−∞
0
0
5
3
0
ex + e−x
2
4. Find the length of the curve parametrized by
x = t2 , y = 4t3 from t = −1 to t = 1.
1. Compute the following integrals or show they
diverge:
ˆ ∞
e−x dx
(a)
ˆ
=
dx
1 + x2
dx
x2 − 9
ln x
dx
x2
x+2
dx
x2 + 4
1