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Section 8.7

Section 8.7. Approximate Integration. 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 = x 0 , x 1 , x 2 , . . . , x n − 1 , x n = b

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Section 8.7

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  1. Section 8.7 Approximate Integration

  2. 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 + iΔx.

  3. THE LEFT AND RIGHT ENDPOINT APPROXIMATIONS Left Endpoint Approximation: Right Endpoint Approximation:

  4. THE MIDPOINT RULE where and

  5. AREA OF A TRAPEZOID Recall the area of a trapezoid is given by

  6. THE TRAPEZOIDAL RULE where and xi = a + iΔx.

  7. 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

  8. PARABOLIC AREA FORMULA The area of the geometric figure below is given by

  9. SIMPSON’S (PARABOLIC) RULE where n is even and

  10. ERROR BOUND FORSIMPSON’S RULE Theorem: Suppose that | f(4)(x) | ≤ K for a≤ x ≤ b. If ES is the error involved in using Simpson’s Rule, then

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