Title: 6.4 Integration with tables and computer algebra systems 6.5 Approximate Integration
16.4 Integration with tables and computer algebra
systems6.5 Approximate Integration
2Tables of Integrals
- A table of 120 integrals, categorized by form,
is provided on the References Pages at the back
of the book. - References to more extensive tables are given in
the textbook. - Integrals do not often occur in exactly the form
listed in a table. - Usually we need to use the Substitution rule or
algebraic manipulation to transform a given
integral into one of the forms in the table. - Example Evaluate
- In the table we have forms involving
- Let u ln x.
- Then using the integral 21 in the table,
3Computer Algebra Systems (CAS)
- Matlab, Mathematica, Maple.
- CAS also can perform substitutions that
transform a given integral into one that occurs
in its stored formulas. - But a hand computation sometimes produces an
indefinite integral in a form that is more
convenient than a machine answer. - Example Evaluate
- Maple and Mathematica give the same answer
-
- If we integrate by hand instead, using the
substitution u x2 5, we get - This is a more convenient form of the answer.
4Can we integrate all continuous functions?
- Most of the functions that we have been dealing
with are what are called elementary functions. - These are the polynomials, rational functions,
exponential functions, logarithmic functions,
trigonometric and inverse trigonometric
functions, - and all functions that can be obtained from these
by the five operations of addition, subtraction,
multiplication, division, and composition. - If f is an elementary function,
- then f is an elementary function,
- but its antiderivative need not be an
elementary function. - Example,
- In fact, the majority of elementary functions
dont have elementary antiderivatives. - How to find definite integrals for those
functions? Approximate!
5Approximating definite integralsRiemann Sums
Recall that the definite integral is defined as a
limit of Riemann sums. A Riemann sum for the
integral of a function f over the interval
a,b is obtained by first dividing the interval
a,b into subintervals and then placing a
rectangle, as shown below, over each subinterval.
The corresponding Riemann sum is the combined
area of the green rectangles. The height of the
rectangle over some given subinterval is the
value of the function f at some point of the
subinterval. This point can be chosen freely.
Taking more division points or subintervals in
the Riemann sums, the approximation of the area
of the domain under the graph of f becomes
better.
6Approximating definite integralsdifferent
choices for the sample points
- Recall that
-
- where xi is any point in the ith subinterval
xi-1,xi. - If xi is chosen to be the left endpoint of the
interval, then xi xi-1 and we have - If xi is chosen to be the right endpoint of the
interval, then xi xi and we have - Ln and Rn are called the left endpoint
approximation and right endpoint approximation ,
respectively.
7Example
First find the exact value using definite
integrals.
Actual area under curve
8Left endpoint approximation
(too low)
9Right endpoint approximation
(too high)
Averaging the right and left endpoint
approximations
(closer to the actual value)
10Averaging the areas of the two rectangles is
the same as taking the area of the trapezoid
above the subinterval.
11Trapezoidal rule
This gives us a better approximation than
either left or right rectangles.
12Can also apply midpoint approximation choose
the midpoint of the subinterval as the sample
point.
The midpoint rule gives a closer approximation
than the trapezoidal rule, but in the opposite
direction.
13Midpoint rule
14Notice that the trapezoidal rule gives us an
answer that has twice as much error as the
midpoint rule, but in the opposite direction.
If we use a weighted average
This is the exact answer!
15This weighted approximation gives us a closer
approximation than the midpoint or trapezoidal
rules.
Midpoint
Trapezoidal
16Simpsons rule
Simpsons rule can also be interpreted as fitting
parabolas to sections of the curve. Simpsons
rule will usually give a very good approximation
with relatively few subintervals.
17Example
18 Error bounds for the approximation methods
Examples of error estimations on the board.