Quick
Review
for
Midterm
1
Sections
covered:
5.10
7.4
5.10
Improper
Integrals
An
improper
integral
is
a
definite
integral
that
has
one
of
the
limits
as
infinity,
or
has
a
discontinuity
in
the
interval
of
interest
(e.g.
an
asymptote).
Type
1:
Infinity
as
one
of
the
limits
!
! ! !" = lim
! !
! ! !"
!
How
to
solve
them?
1.
2.
3.
4.
Replace
the
+/-
with
t.
Solve
the
integral
with
t.
Plug
back
instead
of
t,
and
see
what
value
you
obtain.
If
the
result
is
a
number,
then
the
integral
is
converging,
if
the
limit
doesnt
exist,
then
the
integral
is
divergent.
Type
2:
There
is
a
discontinuity
in
the
interval
(e.g.
an
asymptote)
!
! ! !" = lim
! !
! ! !"
!
How
to
solve
them?
1.
2.
3.
4.
Replace
the
discontinuity
with
t.
Solve
the
integral
with
t.
Plug
back
the
discontinuity
instead
of
t,
and
see
what
value
you
obtain.
If
the
result
is
a
number,
then
the
integral
is
converging,
if
the
limit
doesnt
exist,
then
the
integral
is
divergent.
6.1
More
about
areas
This
section
is
concerned
with
finding
the
area
between
2
curves.
!" ! ! ! ! , !!" !! !"#! !"#$""% ! !"# ! !" !" !"#$%&'( !":
!
! =
[! ! ! ! ]!"
!
Note:
Function
f(x)
HAS
TO
BE
ON
TOP
of
g(x).
If
the
functions
intersect
eachother
on
the
interval
[a,b],
then
split
it
in
2
(or
more)
integrals.
6.2
Volumes
Once
you
know
how
to
find
the
area,
you
can
move
to
finding
the
volume
of
a
solid.
Disk
Method:
If
there
is
a
solid
that
lies
between
a
and
b,
then
you
can
find
the
area
by
looking
at
the
cross-sectional
area
that
is
perpendicular
to
x-axis,
A(x).
If
A(x)
is
a
continuous
function,
then
the
volume
of
the
solid
is:
!
!=
!(!)!"
!
What
this
means
is
simply:
Volume
is
the
integral
of
the
area.
How
to
solve
this?
1. Slice
the
solid
into
very
small
slices.
2. Write
the
area
of
each
slice
as
a
function
of
x.
3. Integrate
that
function
over
the
desired
interval.
Note:
If
it
is
easier
to
slice
along
y-axis,
then
find
A(y)
and
integrate
with
respect
to
dy.
Washer
Method:
The
only
difference
between
this
method
and
the
disk-method,
is
that
in
the
washer
method,
the
area
A(x)
is
the
area
between
2
curves.
The
procedure
is
the
same
as
above.
How
to
solve
this?
1. Slice
the
solid
into
very
small
pieces.
2. Find
the
area
of
each
slice
as
a
function
of
x.
!!"#!!" = !(!"#$% !"#$%&)! !(!"#$% !"#$%&)!
3. Integrate
that
function
over
the
desired
interval.
Cylindrical
Shells
Method:
This
method
is
used
when
you
have
a
function
that
gets
revolved
around
y-axis.
This
method
is
fairly
simple;
just
use
the
following
formula:
!
!=
2!!"(!)!"
!
6.3
Arc
Length
This
section
is
very
simple.
To
find
the
length
of
the
arc
follow
this
procedure:
1. Look
at
your
function
and
determine
which
form
does
it
have.
a. The
curve
has
a
parametric
equation
x
=
f(t)
and
y
=
g(t):
!
!"
!"
!=
!
!"
+
!"
!"
b. The
curve
is
given
with
the
equation
y
=
f(x)
!
!=
1+
!
!"
!"
!"
c. The
curve
is
give
with
the
equation
x
=
f(y)
!
!=
!
!"
1+
!"
!"
6.5
Average
Value
of
a
Function
To
find
the
average
value
of
a
function
over
an
interval,
you
need
to
integrate
it,
and
then
divide
by
the
interval,
i.e.:
!!"#
1
=
!!
!(!)!"
!
6.6
Application
for
Physics
and
Engineering
In
this
section
look
beyond
calculus!
Understand
what
the
question
is
asking
you.
How
do
we
solve
this?
Force:
!!"#$%&' = ! !
!!"#$%& = ! !, !!"! ! !" !! !"#$%&'()$* !" !! !"#$%&
!!"#$$%"# = ! !
Work:
! = !"#$% !"#$%&'(
Hydrostatic
Pressure:
! = ! ! !, !!"! ! !" !! !"#$
How
do
we
solve
this?
1. Write
down
the
general
formula
for
what
the
question
is
asking:
Force,
work,
pressure.
2. Express
each
term
in
that
formula
in
terms
of
x
(if
possible).
3. Integrate
over
the
appropriate
interval.
Its
hard
to
write
a
specific
procedure
on
how
to
solve
these
types
of
problems.
They
require
a
lot
of
practice
to
able
to
set
them
up
properly.
Examples
to
look
at
are
examples
1-5
on
page
466.
Make
sure
you
understand
these
examples,
if
you
dont
make
sure
you
ask
me
to
explain
them
to
you
next
time
we
meet.
7.1-7.4:
Differential
Equations
Again,
as
in
the
previous
sections,
you
need
to
practice
a
lot
in
order
to
be
able
to
properly
set
up
the
equation.
However,
once
you
have
a
differential
equation
set-up,
most
of
the
time
it
would
be
a
separable
equation.
This
is
how
you
solve
them:
1. The
general
form
of
the
separable
differential
equation
is:
!" !(!)
=
(!" !"# !"#$ !!"#$ !" !!")
!" !(!)
2. Move
everything
with
x
on
one
side,
and
everything
with
y
on
another
side.
! ! !" = ! ! !"
3. Integrate
both
sides.
! ! !" =
! ! !"
! ! = ! ! + !
4. Express
the
result
as
y
=
Function(x)
Euler
Method:
Euler
method
is
used
to
approximate
a
solution
to
a
differential
equation
(since,
it
is
not
always
possible
to
solve
a
differential
equation).
This
method
is
really
simple,
but
requires
a
lot
of
tedious
arithmetic
calculations.
Follow
the
procedure
outlined
on
page
504
of
the
textbook.
It
is
very
straight
forward.
