Lecture10 approxating integrals part 1 - Lecture 10 MATH...

Lecture 10: MATH 138-W12-003 January 25, 2012
I) Approximate (Numerical) Integration, Section 7.7
We have learned a plethora of analytical techniques to integrate functions of a single variable.
This allows us to evaluate a lot of integrals exactly, which is great when we can do it. The fact
of the matter is that life usually presents us with integrals that cannot be evaluated. Some of
you might think that this is because we have not thought of the right substitution or the right
method. Indeed, this could be the case sometimes. But there are some integrals where it can be
proven that it is impossible to find an anti-derivative, a couple of examples are
Z
e
- x
2
dx, and
Z
√
1 + x 3 dx.
If you don’t believe me then I encourage you to give it a try and see what you can come up with.
A second reason for wanting to approximate integrals is that in the real world we are rarely
if ever given a continuous function. Instead, we are given data at a given time (or location) and
if we want to find the area below the curve we must find a way to integrate a discrete function.
For example we could have,
f ( x
i
) for x
1
, x
2
, x
3
, · · · , x
n
.
None of our analytical techniques work for this type of problem but they necessarily arise when
we need to integrate.
Both of these problems motivates us wanting to approximate the integrals of functions. This
is done by numerical integration, something that I’ve been told you touched on in MATH 137
and we will expand on here in MATH 138. We begin by reviewing what you’ve already seen
and then present two new methods, one of which is very powerful and what is often used in
real world problems for high efficiency and accuracy.
Review:
If we partition our continuous interval into N bins of length Δ x = ( b - a ) /n for n a positive
integer then we can use Riemann sums to approximate the integral. Three examples that you saw
previously are shown below. First, if the interval is a ≤ x ≤ b we partition it into n subintervals
where the end points are
x
0
= a, x
1
= a + Δ x, x
2
= a + 2Δ x, · · · , x
n - 1
= b - Δ x, x
n
= b.