Lecture 19 – Numerical Integration 1 Area under curve led to Riemann sum which led to FTC. But not every function has a closed-form antiderivative.

2 Using rectangles based on the left endpoint of each subinterval. Using rectangles based on the right endpoints of each subinterval ab ab

3 Using rectangles based on the midpoint of each subinterval. ab

4 Regardless of what determines height:

5 Example 1 Use the midpoint rule to estimate the area from 0 to

Example 2 Compare the three rectangle methods in estimating area from x = 1 to 9 using 4 subintervals. 6 f(x) x

Lecture 20 – More Numerical Integration 7 Instead of rectangles, look at other types easy to compute. Trapezoid Rule: average of Left and Right estimates Area for one trapezoid is (average length of parallel sides) times (width). ababab

Trapezoid Rule is the average of the left and right estimates, so 8 x0x0 x1x1 x n-1 x2x2 xnxn

9 Simpson’s Rule: weighted average of Mid and Trap estimates a – must break into even number of subintervals – areas under quadratic curves – pairs of subintervals form quadratic function b

Simpson’s Rule is the sum of these areas, so 10 Calculate efficiency of estimates with absolute errors, relative errors, and percent error (change decimal of relative to %).

11 Example 3 Use the trapezoid and Simpson’s rules to estimate the integral. 9 1

12 Example 4 Use the M 6, T 6, and S 6 to fill in the table for the given integral. 8 2 RuleEstimateAbsolute Error Relative Error M6M6 T6T6 S6S6