2. Do WebAssign 7.6 #1 - #3 and #5 - #7.
Z
π/2
6.
π/4
sin(s)
p
ds
cos(s)
(3-6) Determine whether the following integrals converge. (You may calculate the limits by appealing
to the dominance of one function over another, or by
l’Hopital’s rule.) If the integral converges, give its
exact value.
Z ∞
x
dx
3.
4
+
x2
1
Z
4.
0
4
1
√ dx
x
7. Given that
value of
R∞
−∞
√
2
e−x dx = π, calculate the exact
Z ∞
2
e−(x−a) /b
−∞
Z
5.
2
∞
1
dx
x ln(x)
Comparing Integrals 1
Understand
Apply
Understand
Apply
Apply
Apply
Synthesize
Z
∞
1
1
dx
x2
Z
∞
(b)
1
∞
(d)
1
Z
(e)
1
Z
Name:
Math 129 - 20
Rb
Use the terminology “improper integral” for a g(x) dx where g(x) is unbounded on [a, b].
Rb
Rc
If g(x) is unbounded at b calculate the integral a g(x) dx, by calculating limc→b− a g(x) dx
Use the terminology “converges” if the result of this limit is finite, and “diverges” if it is infinte.
Rb
If g(x) is unbounded at c, with a < c < b, calculate the integral a g(x) dx, by splitting it into two
Rc
Rb
integrals a Rg(x) dx + c g(x) dx.
∞
Know
that 1 x1p dx converges if
and diverges if
.
R ∞ −ax
and
diverges
if
.
e
dx
converges
if
0
Use algebra, substitutions, and l’Hopital’s Rule to match any rational integrand to x1p for some p.
1. Determine whether the following integrals converge
or diverge. You should calculate the first one using
methods from section 7.6. For each of the following
integrals, you may either make a calculation or a comparison argument as in section 7.7. Include briefly the
main idea of your calculation or argument, without
going into too much detail.
(a)
Section 7.6b and 7.7
February 9, 2016
x2
5x
dx
+ 64
x3
5x
dx
+ 64x
∞
5
dx
x2
(f) Why the fourth integral is the odd one out?
Z
(c)
1
∞
5
dx
x2 + 64
(g) Write one sentence explaining how the integrand of
the the fourth integral is different from the others.
Quiz (Leave this space blank)