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Lecture 19 – Numerical Integration

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.

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Lecture 19 – Numerical Integration

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  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. a b

  4. Regardless of what determines height:

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

  6. Example 2 f(x) Compare the three rectangle methods in estimating area from x = 1 to 9 using 4 subintervals. 3 4 7 1 2 5 6 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, so x1 x2 xn-1 xn x0

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

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

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