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Calculus
Lesson
25
Numerical Integration
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You need to be familiar with both Trapezoidal approximations and
Simpson's Rule for the AP Exam. Other than that, they are not that
important!
The Trapezoidal Rule is an average of the small and large rectangular
approximations.
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Note that the area of a trapezoid is the average length of the bases times the width
so for each section, the area is:
If there were two sections, this would become:
Since 
if
we break the area up into equal parts, we can simplify this way:
Lastly, note that if our interval is from 
to 
that if we break the
interval down into 
intervals,
that each interval size is 
.
Also, since each part will be divided by 2, we can pull out the
1/2. Finally, notice that except for the first and last
sides, each side is counted twice.
Hence, the formula for the trapezoidal rule area is:
Notice the pattern 1, 2, 2, 2, 2, 2, 2, ..., 2, 2, 2, 1
Examples (do each for n=4, 8 and check against fnint: 
Simpson's Rule is similar, but instead of using straight lines between the two points that
border each interval, we use a parabola to connect three points that border two intervals.
Note that there is a parabola that connects any three points.
For Simpson's Rule, the pattern is 1, 4, 2, 4, 2, 4, 2, 4, 2, ..., 2, 4, 2, 4, 2, 4, 1.
So for trapezoid, divide by 2 and the pattern is 1, 2, 2, 2, 2, 2, 2, ..., 2, 2, 2, 1.
For Simpson's, divide by 3 and the pattern is 1, 4, 2, 4, 2, 4, 2, 4, 2, ..., 2, 4, 2, 4, 2, 4, 1.
Also (Simpson's), you must have an odd number of points (an even number of intervals!)
Do the same examples as before and compare them to trap and fnint.
Also note that you get the exact answer for the quadratic with Simpson's!
Euler's Method
Newton's Method was for finding the x, given we want y=0.
For Euler's Method, we want to find the value of y, given that we know the value of x.
This is even more intuitive than Newton's Method! To find the next x, imagine the curve is similar to a line (which is true if the interval is small enough).
Example: 
. We can easily tell
what 
is, but what
is 
?
Let's do it in 2 steps: And in 4 steps:
Eight steps gives 
Sixteen steps gives 
Thirty-two steps gives 
Sixty-four steps gives 
This may seem like a silly thing to do for a function that we can actually evaluate, like the function above, but what about:
,
solve for 
using one
step and four steps.
,
solve for 
using two
steps and four steps.