1. Approximate Integration
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 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