Math 152 summary and review guide
5.1,5.2,6.5
Area and distance, Riemann sum, Definite integral, Average
Write a definite integral as a Riemann sum and vice versa. Interpret
an integral as a (signed) area under a curve, or accumulated distance
given the velocity.
Use formulas for
∑
n
i
=1
i
and
∑
n
i
=1
i
2
to com
pute integrals (p.396). Understand basic and comparison properties
(p.373 and p.375). Average value of
f
over [
a, b
] is
1
b
−
a
integraltext
b
a
f
(
x
)
dx
. By
the Mean Value theorem this average equals
f
(
c
) for some
c
∈
(
a, b
)
if
f
is continuous.
5.3
Fundamental theorem of calculus (FTC)
Roughly, if
F
′
(
x
) =
f
(
x
) then
F
(
x
) =
integraltext
x
a
f
(
t
)
dt
and
integraltext
b
a
f
(
x
)
dx
=
F
(
b
)

F
(
a
). Use with
chain rule to compute
d
dx
integraltext
g
(
x
)
f
(
x
)
h
(
t
)
dt
=
h
(
g
(
x
))
g
′
(
x
)

h
(
f
(
x
))
f
′
(
x
)+
C
.
5.4
Indefinite integrals.
integraltext
f
(
x
)
dx
=
F
(
x
) +
C
means that
F
(
x
)

integraltext
x
a
f
(
t
)
dt
is a constant function, where the constant depends on the
value of
a
. Memorize the table on p.392 except for the formulas in
volving csc
x
. The “Net change theorem” (p.394) is just the FTC.
“Displacement” versus “Distance” (p.395). Power is the time deriv
ative of energy
P
(
t
) =
E
′
(
t
) (p.396).
5.5
Substitution rule
This is the chain rule in reverse (p.401). Look
for the form
integraltext
b
a
f
(
g
(
x
))
g
′
(
x
)
dx
and adjust the limits of integration
(p.404). Symmetric functions sometimes integrate easier (p.405).
6.1, 10.4
Area between curves in Cartesian and polar coordinates
Strategy: Sketch curves, and decide whether to integrate with respect
to
x
or
y
. Find intersection points, and decide where
f
(
x
)
≤
g
(
x
)
to compute
Area
=
integraltext
b
a

f
(
x
)

g
(
x
)

dx
.
Practice polar plotting of
r
=
f
(
θ
), and use
A
=
integraltext
b
a
1
2
r
2
dθ
. Be aware of intersection points that
arise from the facts (
r, θ
) = (

r, θ
+
π
) and (0
, θ
) = (0
, θ
′
) (p.651).
6.2, 6.3
Volume of a solid of revolution by disk, shell and direct
methods
Strategy: Sketch and visualize the solid. Decide between
washers or cylindrical shells. For
y
=
f
(
x
), washers are better around
the
x
axis and cylinders are better around the
y
axis. Find intersec
tion points for limits of integration. For washers around
x
axis, use
V
=
integraltext
b
a
A
(
x
)
dx
=
integraltext
b
a
π
([
f
(
x
)]
2

[
g
(
x
)]
2
)
dx
(p.426).
For cylinders
around
y
axis, use
V
=
integraltext
b
a
A
(
x
)
dx
=
integraltext
b
a
2
πxf
(
x
)
dy
(p.434).
For
other solids, decide on a “slicing strategy”, plot the object so that
the area of the “slice at
x
”,
A
(
x
), is a simple function, and find the
limits of integration for
V
=
integraltext
b
a
A
(
x
)
dx
(p.429).
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 Fall '08
 VIGHEN
 dx, series A

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