NUMERICAL INTEGRATION
Mid-point RULE We will divide the interval [a, b] into n subintervals of equal width, We will denote each of the intervals as follows, where and
Mid-point RULE Then for each interval let be the midpoint of the interval. We then sketch in rectangles for each subinterval with a height of . Here is a graph showing the set up using n=6
Mid-point RULE
Mid-point RULE We can easily find the area for each of these rectangles and so for a general n we get that, Or, upon factoring out we get the general Rectangular Rule:
TRAPEZOIDAL RULE For this rule we will do the same set up as for the Rectangular Rule. We will break up the interval [a, b]into n subintervals of width, Then on each subinterval we will approximate the function with a straight line that is equal to the function values at either endpoint of the interval. Here is a sketch of this case for n=6 .
TRAPEZOIDAL RULE
TRAPEZOIDAL RULE The area of the trapezoid in the interval is given by So, if we use n subintervals the integral is approximately,
TRAPEZOIDAL RULE Simplifying the equation, gives us the Trapezoidal rule as
SIMPSON’S RULE We will again divide up the interval [a, b] into n subintervals. The width of each subinterval is, In the Trapezoidal Rule we approximated the curve with a straight line. For Simpson’s Rule we are going to approximate the function with a quadratic equation
SIMPSON’S RULE
SIMPSON’S RULE Notice that each approximation actually covers two of the subintervals. So for Simpson’s rule we require n to be even. It can be shown that the area under the approximation on the intervals and is,
SIMPSON’S RULE If we use n subintervals the integral is then approximately, The general form of Simpson’s rule is
Examples i xi* f(xi*) 1 1.125 1.42 2 1.375 2.60 3 1.625 4.29 4 1.875 6.59
Trapezoidal Rule Simpson’s Rule i xi f(xi) 1 1.25 1.95 2 1.5 3.38 3 1.75 5.36 8 Simpson’s Rule
Extra practice sums Apply Simpson’s rule to evaluate with n = 6. Evaluate using Simpson’s rule with n = 4. Evaluate using Trapezoidal rule with n = 5.