Arc Length with Vector Functions
In this section well recast an old formula into terms of vector functions. We want to
determine the length of a vector function,
on the interval
.
We actually already know how to do this. Recall that we can write the vecto
Functions of Several Variables
In this section we want to go over some of the basic ideas about functions of more
than one variable.
First, remember that graphs of functions of two variables,
are surfaces in three dimensional space. For example here is th
Hydrostatic Pressure and Force
In this section we are going to submerge a vertical plate in water and we want to know
the force that is exerted on the plate due to the pressure of the water. This force is
often called the hydrostatic force.
There are two
Improper Integrals
In this section we need to take a look at a couple of different kinds of integrals. Both
of these are examples of integrals that are called Improper Integrals.
Lets start with the first kind of improper integrals that were going to take
Integral Test
The last topic that we discussed in the previous section was the harmonic series. In
that discussion we stated that the harmonic series was a divergent series. It is now
time to prove that statement. This proof will also get us started on th
Integrals Involving Quadratics
To this point weve seen quite a few integrals that involve quadratics. A couple of
examples are,
We also saw that integrals involving
,
done with a trig substitution.
and
could be
Notice however that all of these integrals w
Integrals Involving Roots
In this section were going to look at an integration technique that can be useful
for some integrals with roots in them. Weve already seen some integrals with roots in
them. Some can be done quickly with a simple Calculus I subst
Integration Strategy
Weve now seen a fair number of different integration techniques and so we should
probably pause at this point and talk a little bit about a strategy to use for determining
the correct technique to use when faced with an integral.
Ther
Lets start off with this section with a couple of integrals that we should already be
able to do to get us started. First lets take a look at the following.
So, that was simple enough. Now, lets take a look at,
To do this integral well use the following s
More on Sequences
In the previous section we introduced the concept of a sequence and talked about
limits of sequences and the idea of convergence and divergence for a sequence. In
this section we want to take a quick look at some ideas involving sequence
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Estimating the Value of a Series
We have now spent quite a few sections determining the convergence of a series,
however, with the exception of geometric and telescoping series, we have not talked
about finding the value of a series. This is usually a ver
Equations of Planes
In the first section of this chapter we saw a couple of equations of planes. However,
none of those equations had three variables in them and were really extensions of
graphs that we could look at in two dimensions. We would like a mor
Arc Length
In this section we are going to look at computing the arc length of a function.
Because its easy enough to derive the formulas that well use in this section we will
derive one of them and leave the other to you to derive.
We want to determine t
Area with Parametric Equations
In this section we will find a formula for determining the area under a parametric
curve given by the parametric equations,
We will also need to further add in the assumption that the curve is traced out exactly
once as t in
Area with Polar Coordinates
In this section we are going to look at areas enclosed by polar curves. Note as well
that we said enclosed by instead of under as we typically have in these problems.
These problems work a little differently in polar coordinate
Binomial Series
In this final section of this chapter we are going to look at another series
representation for a function. Before we do this lets first recall the following
theorem.
Binomial Theorem
If n is any positive integer then,
where,
This is usefu
Center of Mass
In this section we are going to find the center of mass or centroid of a thin plate with
uniform density . The center of mass or centroid of a region is the point in which the
region will be perfectly balanced horizontally if suspended from
Comparison Test for Improper Integrals
Now that weve seen how to actually compute improper integrals we need to address
one more topic about them. Often we arent concerned with the actual value of these
integrals. Instead we might only be interested in wh
Comparison Test / Limit Comparison Test
In the previous section we saw how to relate a series to an improper integral to
determine the convergence of a series. While the integral test is a nice test, it does
force us to do improper integrals which arent a
Cross Product
In this final section of this chapter we will look at the cross product of two vectors.
We should note that the cross product requires both of the vectors to be three
dimensional vectors.
Also, before getting into how to compute these we sho
Curvature
In this section we want to briefly discuss the curvature of a smooth curve (recall that
for a smooth curve we require
is continuous and
). The curvature measures how fast a curve is changing direction at a given
point.
There are several formulas
Cylindrical Coordinates
As with two dimensional space the standard
coordinate
system is called the Cartesian coordinate system. In the last two sections of this
chapter well be looking at some alternate coordinates systems for three dimensional
space.
Wel
Dot Product
The next topic for discussion is that of the dot product. Lets jump right into the
definition of the dot product. Given the two vectors
and
the dot product is,
(1)
Sometimes the dot product is called the scalar product. The dot product is also
Equations of Lines
In this section we need to take a look at the equation of a line in
. As we saw
in the previous section the equation
does not describe
a line in
, instead it describes a plane. This doesnt mean however that we
cant write down an equatio
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