 # Numerical Integration

## Presentation on theme: "Numerical Integration"— Presentation transcript:

Numerical Integration

Definite Integrals

NUMERICAL INTEGRATION
Riemann Sum Use decompositions of the type General kth subinterval:

RULES TO SELECT POINTS Riemann Sum Left Rule

RULES TO SELECT POINTS Riemann Sum Right Rule

RULES TO SELECT POINTS Riemann Sum Midpoint Rule

RULES TO SELECT POINTS Left Approximation LEFT(n) =
Right Approximation RIGHT(n) =

Midpoint Approximation
RULES TO SELECT POINTS Midpoint Approximation MID(n) =

PROPERTIES Property If f is increasing, LEFT(n) RIGHT(n)

PROPERTIES

PROPERTIES Property For any function,

PROPERTIES Property If f is increasing, Hence

If f is increasing or decreasing:
PROPERTIES Property If f is increasing or decreasing:

CONCAVITY Recall The graph of a function f is concave up, if the graph lies above any of its tangent line.

MIDPOINT APPROXIMATIONS
MID(n) =

MIDPOINT APPROXIMATIONS
The two blue areas on the left are the same. The blue polygon in the middle is contained in the domain under the concave-up curve. MID(n)

MIDPOINT APPROXIMATIONS
If the function f takes positive values, and if the graph of f is concave-up MID(n)

MIDPOINT APPROXIMATIONS
If the function f takes positive values, and if the graph of f is concave-down MID(n)

TRAPEZOIDAL APPROXIMATIONS
LEFT(n) rectangle RIGHT(n) rectangle TRAP(n) polygon

TRAPEZOIDAL APPROXIMATIONS
If the function f takes positive values and is concave-up TRAP(n) polygon

COMPARING APPROXIMATIONS
Example f The graph of a function f is increasing and concave up. a b Arrange the various numerical approximations of the integral into an increasing order.

COMPARING APPROXIMATIONS
Example f Because f is increasing, a b Because f is positive and concave-up,

COMPARING APPROXIMATIONS
Example f Because f is increasing and concave-up, a b

COMPARING APPROXIMATIONS
Example f Because f is increasing and concave-up, a b

SUMMARY Left Approximation LEFT(n) = Right Approximation RIGHT(n) =

SUMMARY Midpoint Approximation MID(n) = Trapezoidal Approximation

SIMPSON’S APPROXIMATION
In many cases, Simpson’s Approximation gives best results.