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Clicker Question 1. What is ? A. x tan( x 2 ) + C B. ½ (sec( x 2 ) + tan( x 2 )) + C C. ln |sec( x 2 ) + tan( x 2 )| + C D. ½ tan( x 2 ) + C E. ½ ln |sec( x 2 ) + tan( x 2 )| + C. Clicker Question 2. What is ? A. x 2 tan(x) + C

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clicker question 1
Clicker Question 1
  • What is ?
    • A. x tan(x2) + C
    • B. ½ (sec(x2) + tan(x2)) + C
    • C. ln |sec(x2) + tan(x2)| + C
    • D. ½ tan(x2) + C
    • E. ½ ln |sec(x2) + tan(x2)| + C
clicker question 2
Clicker Question 2
  • What is ?
    • A. x2 tan(x) + C
    • B. x tan(x) + sec2(x) + C
    • C. x tan(x) – ln |sec(x)| + C
    • D. ½ x2 tan(x) + C
    • E. x tan(x) + ln |sec(x)| + C
numerical or approximate integration 2 14 14
Numerical (or Approximate) Integration (2/14/14)
  • 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.
getting good approximations
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
simpson s rule
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).
  • Can get Simpson directly from the data using(1 + 4 + 2 + 4 + 2 +…+ 2 + 4 + 1) / 6
clicker question 3
Clicker Question 3
  • Suppose a definite integral has the following estimates for 10 subdivisions of the interval:Left-hand sum: 12.5Right-hand sum: 16.5Midpoint Rule sum: 13.0What is the Simpson’s Rule estimate?
  • A. 13.5 B. 13.0 C. 13.75 D. 14.0 E. 13.25
assignment for monday
Assignment for Monday
  • Finish up Hand-in #2 (due Monday in class)
  • Section 7.7 goes into much more detail on approximate integration than we need to, and defines Simpson’s Rule slightly differently, 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 (i.e., use 9 data points) both using the weighted average of Trap and Mid and the (1-4-2-4-2-4-2-4-1) / 6 method.