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

Using rectangles based on the left endpoint of each subinterval. Using rectangles based on the right endpoints of each subinterval a b a b

Using rectangles based on the midpoint of each subinterval.

Regardless of what determines height:

Example 1 Use the midpoint rule to estimate the area from 0 to 120. 8 6 4 2 120

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

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).

Trapezoid Rule is the average of the left and right estimates, so x0 x1 x2 xn-1 xn

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

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 %).

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

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