integration by parts:
or
The biggest hint is simply that this integration technique, like all integration techniques, is really just a differentiation technique in reverse.
Integration by parts may need to be carried out multiple times: sometimes the idea is to simplify the integral each time, until a really simple one arises allowing us to calculate the final solution (e.g. Example 6, p. 507). Sometimes it's something of a trick: we compute the integral multiple times in order to return to the original integral, allowing us to solve an equation for the original integral (e.g. Example 4, p. 506).
We encounter integration by parts, which is a way of turning one integral into another that we prefer to integrate by using the product rule in reverse. Similarly, and as mentioned in the text, substitution is really just the chain rule in reverse. Integration and differentiation are inverse processes. One of the differences, however, is that whereas the derivative of a function is unique, the indefinite integral of a function represents a class of functions.
You should recall the definitions of the trig functions (in terms of 
and
).
Trigonometric identities are essentially theorems with well-known proofs. Among the most important are the following:
-
(Pythagorean theorem). If we divide this by either
or 
,
we get a new identity involving 
and 
, or 
and 
. -
(double angle formula), from which we can derive a formula for
by
using the Pythagorean identity. -
(half-angle formula), from which we can derive a formula for
by
using the Pythagorean identity:
-
You might wonder what's so exciting about these types of integrals: it turns out that they occur in an essential way in an important technique called Fourier analysis. The idea is that we can approximate relatively arbitrary functions in terms of trig functions, but we won't go into that now (hints of it in problem #66, p. 517, however!).
This section might be considered ``archaic'' by some: it's full of techniques which we really don't need much anymore, since, if we're confronted by some hairy integral of this class we would be able to get the solution most easily by asking your TI calculator to solve it...!
One of the important lessons of this section is the usefulness of trigonometric identities, many of which you've undoubtably forgotten. Take this opportunity to remember the relationship between tan, sec, sin, and cos, etc., and their derivatives.
Remember your trig identities!
Note that you have to pay attention to the interval on which you make the substitution.
Trig substitution is defined as a method of ``inverse substitution'': rather than replace some complex function of the dummy variable with a new variable u, e.g.
we replace the dummy variable of integration x by some complex function of u! Hence, we might try
and hope that things turn out nicer.
In particular, this section demonstrates that root functions can sometimes be eliminated using this technique.
-
Midpoint rule
where
and
the midpoint of the
subinterval 
. -
Trapezoidal rule
where
and
-
Simpson's rule (1-4-2-4-2 step)
where
and
Error Bounds
-
For the Trapezoidal Rule:
where K is an upper-bound for
on [a,b]. -
For the Midpoint Rule:
where K is an upper-bound for
on [a,b]. -
For Simpson's Rule:
where K is an upper-bound for
on [a,b].
Upper-bounds might be determined algebraically, estimated graphically, or derived from max/min considerations. Note that K is different in the different formulas.
Recall the definitions of the Riemann sum: the sum
where 
, named after Bernhard Riemann (1826-1866), a
student of Gauss.
In general
- These approximation methods get better as the subintervals get smaller (that is, as n gets larger).
- The errors of the trapezoidal and midpoint methods are oppositive in sign, and differ by a factor of two (midpoint is better, in general).
-
Simpson's rules requires an even number of subintervals, and can be written as
a weighted sum of the trapezoidal and midpoint methods:
There are two major motivations for approximate integration:
-
has no ``closed-form'' solution (can't find an anti-derivative); and
-
There is no f(x)! We're working with data pairs
rather than
some known function, as if frequently the case in industry.
which approximates f,
or fits the data pairs , and hope that
The choices for are usually step-functions (Left, Right, and Midpoint
Rectangle rules), or continuous but non-differentiable functions (Trapezoidal
and Simpson's rule). Other (better!) rules use continuous and smooth
functions.
The trapezoidal rule is simply the average of the left and right rectangle rules, a primitive improvement on them both. It is also equivalent to adding up the area under the trapezoids created by connecting left and right endpoints of the curve and then dropping the ends down to the x-axis.
The midpoint rule is created on the hope that we can avoid extremes at the left and right endpoints. It is another simple rectangle rule, but evidently generally better than the trapezoidal rule.
Simpson's rule is derived using ``best-fitting'' parabolas, rather than straight line segments. It is considered a very good method in general.
All the error bounds rely on having a bound on a higher derivative of the function. This may not be easy to obtain.
The arc length formula. If f' is continuous on [a,b], then the
length of the curve y=f(x), 
, is
The arc length function. If smooth curve C has an equation
y=f(x), , let s(x) be the distance along C from the initial
point 
to the point Q(x,f(x)). Then
(note the change in the dummy variable of integration).
Notice how the formulas change if integrating along the y axis.
Archimedes got the whole ball started by computing the value of 
by
approximating a circle by a regular polygon, both inside and out. He thus
squeezed the value of down to between 
.
He thus approximated a smooth curve by a bunch of line segments, a process
which we continue here. As the line segments get finer and finer, the
approximation gets better and better, until, in the limit, the approximation
becomes exact. We can imagine that we're using an approximation
to f, and hoping that the approximation of the arc length of gets
better and better as the line segments become shorter and shorter.
We do invoke the Mean Value Theorem at some point. You might want to remind yourself of how that works!
The surface area S of the surface obtained by rotating the curve
y=f(x), with f positive and with a continuous derivative, ,
about the x axis is defined as
Writing this formula using the notation given for arc length in section 9.1, we find that
which we can think of S as the sum of belts of rectangular area, of thickness
ds and circumference 
.
parametric equations: both x and y are functions of another variable (the parameter) t:
As t varies over its domain, x and y wander about in the plane, creating a parametric curve. As t ranges from a to b, the curve moves from initial point (f(a),g(a)) to terminal point (f(b),g(b)).
cycloid: the curve traced out by a point on the perimeter of a circle as the circle rolls along a straight line. The cycloid figures heavily in the brachistochrone problem (shortest time) and the tautochrone problem (equal time).
We now relax, and allow curves that fail the vertical line test, thinking of them as trajectories or pathways of a particle as it moves in time. Time is hence a ``parameter'' which locates the particle at a particular point in the plane.
Parametric curves are terribly important for computer graphics, as computer gamers can well imagine. As mentioned in the text, the letters in a laser printer may well be drawn as parametric curves (Bezier curves).
The cycloid is an important example of a parametric curve, which proves to be the solution to both the brachistochrone and the tautochrone problems.
It is important to distinguish between the derivatives with respect to the parameter t and with respect to x: in general
One reflects the time rate of change in y; the other indicates the tangential rate of change in the curve of y=f(x).
Suppose that we can write our parametric curves as y=F(x):
For example,
So what's the problem with this representation which suppresses the t?
The problem is that the same curve is used over and over as the path for the
parametric motion. The point turns around whenever 
: at
those times, the motion is stopped, and the point is turning around on the
parabola.
If a curve C is described by the parametric equations x=f(t), y=g(t),
, where 
and 
are continuous on
and C is traversed exactly once as t increases from
to 
, then the length of C is
If a curve C is described by the parametric equations x=f(t), y=g(t),
, where and are continuous on
, 
, then the surface area of the surface
obtained by rotating C about the x-axis is given by S, where
This is just a change of variables problem!
Yes, that's right: this is just about change of variable. We start with our old
formulas: for example, if f' is continuous on [a,b], then the length of the
curve y=f(x), , is
Now, suppose that x and y are given parametrically, as x=f(t) and y=g(t). Then
where 
and 
. (Note: since 
must
exist, x is travelling from left to right or from right to left, and doesn't
stop; that is, 
).
The polar coordinate system (introduced by Newton) is an alternative to
the Cartesian coordinate system (named after Descartes) in which every point in
the plane is expressed by its distance and direction (angle) from the origin,
called the pole. The polar axis plays the role formerly played by
the positive x-axis. The polar
coordinates are often given as 
, where
- r is the distance from the pole, and
-
is the angle between the polar axis and the ray passing through the
point of interest.
The graph of a polar equation contains all points P that have at least
one polar representation whose coordinates solve the equation.
Given r and it's easy to find the corresponding x and y:
It's not quite so easy to go in the opposite direction, unless we restrict r
and :
gives a unique representation in polar coordinates (except for the pole, which
has equation r=0 regardless of the value of ).
As a convenience we let r take negative values: so
In any event, we see that, by contrast with the Cartesian coordinate system, the polar coordinate system allows points to have multiple representations.
To find the Cartesian coordinates from the polar coordinates, we can use the equations
Be careful, however, as there are two values of that solve these
equations in each interval of 
, and one must choose the proper value
based on the quadrant in which (x,y) lies....
To plot a polar equation, we may substitute into the equations for x and y:
and treat this as an ordinary parametric equation.
The Cartesian coordinate system is very nice for working with horizontal and vertical lines. It is very easy to represent them in Cartesian coordinates:
- vertical lines have the form x=a, and
- horizontal lines have the form y=b.
In fact, the coordinate axes are, in fact, examples of each of these (x=0 and y=0).
Similarly, the objects most easily represented in polar coordinates would have the form
- r=a, or
-
The first is the form of a circle centered at the origin, whereas the second is the equation of a ray shooting out from the origin, or pole. Notice that when one of these circles and one of these rays intersect, they do so at right angles (just like the coordinate axes in the Cartesian coordinate system).
If you look at the drumheads shown on page 726, you'll see why this representation might be very useful: there is a lot of circular symmetry, and action along lines eminating from the origin. Situations of this type are often better treated in polar, rather than Cartesian, coordinates.