Final Review
Edward Carter
May 13, 2005
1
Parametric Equations and Polar Coordinates
1.1
Things to Know
•
If
dx
dt
= 0,
dy
dx
=
dy
dt
dx
dt
and
d
2
y
dx
2
=
d
dt
(
dy
dx
)
dx
dt
. Higher derivatives are computed in a similar manner.
•
For a parametric curve
x
=
f
(
t
),
y
=
g
(
t
),
a
≤
t
≤
b
, the area under the curve is given by
b
a
g
(
t
)
f
(
t
)
dt
.
One thing to note about this formula is that it gives a number that depends on
how many times
your curve wraps around the area you’re trying to find, as well as
which way
(that is, clockwise or
counterclockwise).
•
The arclength of a parametric curve where
a
≤
t
≤
b
is given by
b
a
dy
dt
2
+
dx
dt
2
dt.
If you’re trying to find the length of a closed curve, make sure you aren’t wrapping around multiple
times.
•
The surface area obtained by rotating a parametric curve with
a
≤
t
≤
b
is given by
b
a
2
πy
dx
dt
2
+
dy
dt
2
dt.
Again, make sure you aren’t wrapping around multiple times in the case of a closed curve. Here, there’s
also the danger that parts of your curve are symmetric about the
x
axis.
That can also cause this
formula to give you numbers that aren’t what you’re looking for.
•
To convert between polar to cartesian,
x
=
r
cos
θ
and
y
= sin
θ
. Use these formulas most of the time.
•
To convert from cartesian to polar,
r
2
=
x
2
+
y
2
and tan
θ
=
y
x
. Use these formulas if you are given
x
and
y
in terms of a third variable such as
t
and you want to obtain
r
and
θ
in terms of
t
.
•
Polar area, for a curve where
a
≤
θ
≤
b
, is given by
b
a
1
2
r
2
dθ
.
•
Polar arclength, for a curve where
a
≤
θ
≤
b
, is given by
b
a
r
2
+
dr
dθ
2
dθ.
1
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1.2
Problems
1. (Practice Midterm 1 for Midterm 1) Describe the planar curve given parametrically by
x
= cosh
t
and
y
= sinh
t
using a relation between
x
and
y
.
2. (Practice Midterm 1 for Midterm 1) Sketch the curve
r
= 2 + 3 cos
θ
and find the area enclosed by its
inner loop.
3. (Practice Midterm 2 for Midterm 1) Sketch the curve
r
= cos
2
θ
and find the area enclosed by it.
4. (Chapter 10 Review) Sketch the curve
x
= tan
t
,
y
= cot
t
by eliminating
t
.
5. (Chapter 10 Review) Sketch the curve
x
= 1 +
e
2
t
,
y
=
e
t
by eliminating
t
.
6. (Chapter 10 Review) Sketch the curve
r
= 1

3 sin
θ
and find the area enclosed by its inner loop.
7. (Chapter 10 Review) Find the points of intersection of the curves
r
= cot
θ
and
r
= 2 cos
θ
.
8. (Chapter 10 Review) Sketch the curve
r
= sin 4
θ
and find the area it encloses.
9. Sketch the curve
r
2
= sin 2
θ
and find the area it encloses.
10. Sketch the curve
r
2
θ
= 1.
2
Vectors, Lines, and Planes
2.1
Things to Know
•
a
·
a
=

a

2
.
•
a
·
(
b
+
c
) =
a
·
b
+
a
·
c
.
•
0
·
a
= 0.
•
a
·
b
=
b
·
a
.
•
(
c
a
)
·
b
=
c
(
a
·
b
) =
a
·
(
c
b
).
•
If
a
and
b
are both nonzero and
θ
is the angle between them, then
a
·
b
=

a

b

cos
θ
. This fact is used
to derive formulas for the scalar and vector projections, which you should also know.
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 Fall '08
 Lanzara
 Physics, Derivative, Vector Calculus

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