1. CHAPTER 4 SECTION 4.6 NUMERICAL INTEGRATION

3. Theorem 4.16 The Trapezoidal Rule

4. Trapezoidal Rule
Instead of calculating approximation rectangles we will use trapezoids
More accuracy
Area of a trapezoid a b •
Which dimension is the h?
Which is the b1 and the b2 b1 b2 h

5. Averaging the areas of
the two rectangles is the
same as taking the area
of the trapezoid above
the subinterval.

6. Trapezoidal Rule Trapezoidal rule approximates the integral f(xi-1)dx
f(xi)
f(xi-1)
Trapezoidal rule approximates the integral

7. Approximate using the trapezoidal rule and n = 4.

8. Approximate using the trapezoidal rule and n = 4.
Actual area:

9. Approximating with the Trapezoidal Rule
Use the Trapezoidal Rule to approximate
y = sin x 1
Four subintervals
y = sin x 1
Eight subintervals

10. Snidly Fizbane Simpson
Simpson's Rule
As before, we divide the interval into n parts
n must be even
Instead of straight lines we draw parabolas through each group of three consecutive points
This approximates the original curve for finding definite integral – formula shown below
Snidly Fizbane Simpson a b •

11. Theorem 4.17 Integral of p(x) =Ax2 + Bx + C

12. Theorem 4.18 Simpson's Rule (n is even)

13. Approximate using Simpson's Rule.
Use Simpson's Rule to approximate. Compare the results for
n = 4 and n = 8.

14. Theorem 4.19 Errors in the Trapezoidal Rule and Simpson's Rule

15. The Approximate Error in the Trapezoidal Rule:
Determine a value n such that the Trapezoidal Rule will approximate the value of with an error less than 0.01.
First find the second derivative of

16. The maximum of occurs at x = 0 (see the graph below)
(Note: the 1st derivative test of
gives
, which would be at x = 0.

17. Thus were one to pick n ≥ 3, one would have an error less than 0.01.
And at x = 0, f '' = 1
The error is thus
Thus were one to pick n ≥ 3, one would have an error less than 0.01.

19. ½ (SUM OF BASES) X height
NOTE: THEY SIMPLY USED
½ (SUM OF BASES) X height
THE FORMULA FOR THE AREA OF A TRAPAZOID!!!!!!

20. John is stressed out. He's got all kinds of projects going on, and they're all falling due around the same time. He's trying to stay calm, but despite all of his yoga techniques, relaxation methods, and sudden switch to decaffeinated coffee, his nerves are still on edge. Forget the 50-page paper due in two weeks and the 90-minute oral presentation next Monday morning; he still hasn't done laundry in weeks, and his stink is beginning to turn heads. Because of all the stress (and possibly due to his lack of bathing), John's started to lose his hair.
Below is a chart representing John's rate of hair loss (in follicles per day) on various days throughout a two-week period. Use 6 trapezoids to approximate John's total hair loss over that traumatic 14-day period.

21. You may be tempted to use the Trapezoid Rule, but you can't use that handy formula, because not all of the trapezoids have the same width. Between day 1 and 4, for example, the width of the approximating trapezoid will be 3, but the next trapezoid will be 6 – 4 = 2 units wide. Therefore, you need to calculate each trapezoid's area separately, knowing that the area of a trapezoid is equal to one-half of the product of the width of the trapezoid and the sum of the bases:

24. WHY DID THEY USE THE TRAPAZOID RULE???!!!!