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APPROXIMATE INTEGRATION

APPROXIMATE INTEGRATION. Example 1 a. The approximation is illustrated here. APPROXIMATE INTEGRATION. Example 1 b. The midpoints of the five subintervals are: 1.1, 1.3, 1.5, 1.7, 1.9. OBSERVATION 1. In all the methods. we get more accurate approximations when we increase n .

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APPROXIMATE INTEGRATION

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  1. APPROXIMATE INTEGRATION Example 1 a • The approximation is illustrated here.

  2. APPROXIMATE INTEGRATION Example 1 b • The midpoints of the five subintervals are: 1.1, 1.3, 1.5, 1.7, 1.9

  3. OBSERVATION 1 • In all the methods. we get more accurate approximations when we increase n. • However, very large values of n result in so many arithmetic operations that we have to beware of accumulated round-off error.

  4. OBSERVATION 2 • The errors in the left and right endpoint approximations are: • Opposite in sign • Appear to decrease by a factor of about 2 when we double the value of n

  5. OBSERVATION 3 • The Trapezoidal and Midpoint Rules are much more accurate than the endpoint approximations.

  6. OBSERVATION 4 • The errors in the Trapezoidal and Midpoint Rules are: • Opposite in sign • Appear to decrease by a factor of about 4 when we double the value of n

  7. OBSERVATION 5 • The size of the error in the Midpoint Rule is about half that in the Trapezoidal Rule.

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