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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 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 = x0, x1, x2, . . . , xn− 1, xn = b where xi = a + iΔx.
THE LEFT AND RIGHT ENDPOINT APPROXIMATIONS Left Endpoint Approximation: Right Endpoint Approximation:
THE MIDPOINT RULE where and
AREA OF A TRAPEZOID Recall the area of a trapezoid is given by
THE TRAPEZOIDAL RULE where and xi = a + iΔx.
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
PARABOLIC AREA FORMULA The area of the geometric figure below is given by
SIMPSON’S (PARABOLIC) RULE where n is even and
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