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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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