Approximate Integration

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Section 8.7

• Approximate Integration

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DIVIDING THE INTERVAL a, b
In order to approximate the integral we must
first partition the interval a, b into n equal
subintervals of with with endpoints a x0, x1,
x2, . . . , xn - 1, xn b where xi a i?x.
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THE LEFT AND RIGHT ENDPOINT APPROXIMATIONS
Left Endpoint Approximation
Right Endpoint Approximation
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THE MIDPOINT RULE
where and
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AREA OF A TRAPEZOID
Recall the area of a trapezoid is given by
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THE TRAPEZOIDAL RULE
where and xi a i?x.
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ERROR BOUNDS FOR THE TRAPEZOIDAL AND MIDPOINT
RULES
Theorem Suppose f ?(x) K for a x
b. If ET and EM are the errors in the
Trapezoidal and Midpoint Rules, respectively, then
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PARABOLIC AREA FORMULA
The area of the geometric figure below is given by
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SIMPSONS (PARABOLIC) RULE
where n is even and
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ERROR BOUND FORSIMPSONS RULE
Theorem Suppose that f (4)(x) K for a
x b. If ES is the error involved in using
Simpsons Rule, then