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6.4 Integration with tables and computer algebra systems 6.5 Approximate Integration. Tables 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.
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.
then f ’ is an elementary function,
but its antiderivative need not be an elementary function.
In fact, the majority of elementary functions don’t have elementary antiderivatives.
How to find definite integrals for those functions? Approximate!
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.
where xi* is any point in the ith subinterval [xi-1,xi].
First find the exact value using definite integrals.
Actual area under curve:
Left endpoint approximation:
Right endpoint approximation:
Averaging the right and left endpoint approximations:
(closer to the actual value)
the two rectangles is the
same as taking the area
of the trapezoid above
This gives us a better approximation
than either left or right rectangles.
Can also apply
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.
(higher than the
(lower than the
Notice 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!
This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules.
Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve.
Simpson’s rule will usually give a very good approximation with relatively few subintervals.
Examples of error estimations on the board.