1. Clicker Question 1 What is ? (Hint: u-sub) – A. ln(x – 2) + C – B. x – x 2 + C – C. x + ln(x – 2) + C – D. x + 2 ln(x – 2) + C – E. 1 / (x – 2) 2 + C

2. Clicker Question 2 The general anti-derivative of a function is F(x) = sin 2 (x) + ln(x) – 3ln(x + 1) + C. Which of the following is another way to express the same function? – A. sin 2 (x) + ln(1 – 2x) + C – B. –cos 2 (x) + ln(x / (x + 1) 3 ) + C – C. cos 2 (x) + ln(x / (x + 1) 3 ) + C – D. –cos 2 (x) + ln(x(x + 1) 3 ) + C – E. –cos 2 (x) + ln(x / (3(x + 1))) + C

3. Numerical (or Approximate) Integration (9/27/13) To evaluate a definite integral, we always hope to apply the Fundamental Theorem by finding an antiderivative and then evaluating it at the endpoints. But this isn’t always possible or practical. Another option always available is to do numerical integration to approximate the answer. Here we use Riemann sums.

4. Getting Good Approximations In general, the more data you use (i.e., the more rectangles you measure), the better the estimate. Methods (in increasing order of accuracy for a fixed number of rectangles) – Left or right-hand endpoints – Trapezoid Rule (average of the above) – Midpoint Rule – Simpson’s Rule

5. Simpson’s Rule It turns out that the Midpoint Rule is about twice as accurate as the Trapezoid Rule and the errors are in opposite directions. Hence we form a “weighted average”: Simpson’s Rule = (2 Midpoint + Trapezoid) / 3 Simpson’s Rule is in general extremely accurate even for small n (i.e., few rectangles).

6. Clicker Question 3 Suppose a definite integral has the following estimates for 10 subdivisions of the interval: Left-hand sum: 12.5 Right-hand sum: 16.5 Midpoint Rule sum: 13.0 What is the Simpson’s Rule estimate? A. 13.5 B. 13.0 C. 13.75 D. 14.0 E. 13.25

7. Going Directly to Simpson You can compute Simpson’s Rule directly from the data points, by, for each subdivision, giving a weight of 1 to each of the two endpoints and a weight of 4 to the mid-point, and then dividing by 1+ 4 + 1 = 6. So the direct Simpson formula becomes (subinterval length)/6 [f (far left endpoint) + 4(f (first midpoint)) + 2 f (first internal endpoint) +....+ 4(f (last midpoint) + f (far right endpoint)]. Or, more briefly: S =  x / 6 (1 + 4 + 2 + 4 + 2 +.......+ 2 + 4 + 1)

8. Assignment for Monday Section 7.7 goes into much more detail on approximate integration than we need to, so just work from our notes please. Do Exercise 3 on page 516 (note: no known way to use FTC on this one), but also compute the Simpson’s Rule estimate with n = 4 both using the weighted average of Trap and Mid and the (1-4-2-4-2-4-2-4-1) / 6 method.