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Numerical Integration
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Definite Integrals
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NUMERICAL INTEGRATION
Riemann Sum Use decompositions of the type General kth subinterval:
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RULES TO SELECT POINTS Riemann Sum Left Rule
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RULES TO SELECT POINTS Riemann Sum Right Rule
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RULES TO SELECT POINTS Riemann Sum Midpoint Rule
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RULES TO SELECT POINTS Left Approximation LEFT(n) =
Right Approximation RIGHT(n) =
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Midpoint Approximation
RULES TO SELECT POINTS Midpoint Approximation MID(n) =
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PROPERTIES Property If f is increasing, LEFT(n) RIGHT(n)
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PROPERTIES
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PROPERTIES Property For any function,
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PROPERTIES Property If f is increasing, Hence
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If f is increasing or decreasing:
PROPERTIES Property If f is increasing or decreasing:
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CONCAVITY Recall The graph of a function f is concave up, if the graph lies above any of its tangent line.
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MIDPOINT APPROXIMATIONS
MID(n) =
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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)
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MIDPOINT APPROXIMATIONS
If the function f takes positive values, and if the graph of f is concave-up MID(n)
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MIDPOINT APPROXIMATIONS
If the function f takes positive values, and if the graph of f is concave-down MID(n)
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TRAPEZOIDAL APPROXIMATIONS
LEFT(n) rectangle RIGHT(n) rectangle TRAP(n) polygon
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TRAPEZOIDAL APPROXIMATIONS
If the function f takes positive values and is concave-up TRAP(n) polygon
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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.
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COMPARING APPROXIMATIONS
Example f Because f is increasing, a b Because f is positive and concave-up,
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COMPARING APPROXIMATIONS
Example f Because f is increasing and concave-up, a b
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COMPARING APPROXIMATIONS
Example f Because f is increasing and concave-up, a b
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SUMMARY Left Approximation LEFT(n) = Right Approximation RIGHT(n) =
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SUMMARY Midpoint Approximation MID(n) = Trapezoidal Approximation
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SIMPSON’S APPROXIMATION
In many cases, Simpson’s Approximation gives best results.
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