Approximate Integration 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 FOR SIMPSON’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