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.