When an integral contains a quadratic expression
, you can sometimes
simplify the integrand by completing the square. This eliminates the middle term of
the quadratic; the resulting integral can then be computed using (e.g.) trig substitution.
Example. Co
Partial fractions is the opposite of adding fractions over a common denominator. It
applies to integrals of the form
The idea is to break
(A function of the form
a rational function.)
into a sum of smaller terms which are easier to integrate.
, where
and
Trig substitution reduces certain integrals to integrals of trig functions. The idea is to
match the given integral against one of the following trig identities:
If the integral contains an expression of the form
based on the first identity:
.
If the inte
How do you find the area of a region bounded by two curves? I'll consider two cases.
Suppose the region is bounded above and below by the two curves
and on the sides by
and
and
,
.
I divide the region up into n vertical rectangles. A typical vertical rect
If u and v are functions of x, the Product Rule says that
Integrate both sides:
This is the integration by parts formula. The integral on the left corresponds to the
integral you're trying to do. Parts replaces it with a term that doesn't need integration
The work required to raise a weight of P pounds a distance of y feet is
footpounds. (In m-k-s units, one would say that a force of k newtons exerted over a
distance of y feet does
newton-meters, or joules, of work.)
Example. If a 100 pound weight is lifte
Start with an area - a planar region - which you can imagine as a piece of
cardboard. The cardboard is attached by one edge to a stick (the axis of revolution).
As you spin the stick, the area revolves and sweeps out a region in space.
The problem is to f
L'Hopital's Rule is a method for computing a limit of the form
c can be a number,
, or
. The conditions for applying it are:
1.
The functions f and g are differentiable in an open interval containing c. (c may
also be an endpoint of the open interval, if
For trig integrals involving powers of sines and cosines, there are two important cases:
1.
The integral contains an odd power of sine or cosine.
2.
The integral contains only even powers of sines and cosines.
I will look at the odd power case first. It t
Roughly speaking, an integral
is improper if:
1.
One of the limits is infinite.
2.
The integrand "blows up" somewhere on the interval of integration.
For example,
are improper because they have infinite limits of integration.
are improper because the inte