1. Lecture 19 – Numerical Integration
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 a b a b

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

4. Regardless of what determines height:

5. Example 1 Use the midpoint rule to estimate the area from 0 to 120. 86 4 2
120

6. Example 2
Compare the three rectangle methods in estimating area from x = 1 to 9 using 4 subintervals.
f(x) 1 2 3 4 5 6 7 8 9 x 1 9

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

8. Trapezoid Rule is the average of the left and right estimates, sox0 x1 x2
xn-1 xn

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

10. Simpson’s Rule is the sum of these areas, so
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. 1 9

12. Example 4 Rule Estimate Absolute Error Relative
Use the M6, T6, and S6 to fill in the table for the given integral. 2 8
Rule
Estimate
Absolute
Error
Relative M6 T6 S6