| 41. | Note that the quadrature rule does not include the endpoints, where we have assumed that the integrand goes to zero.
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| 42. | :: The line integral along a closed curve need not be zero unless the vector integrand is ir-rotational.
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| 43. | But what about the boundaries of the integral, since a definite integral equals the average integrand times the boundary difference:
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| 44. | To show the second integral is small for ? small, it suffices to show that the integrand is uniformly bounded.
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| 45. | They are encoded in the positive geometry of the amplituhedron, via the singularity structure of the integrand for scattering amplitudes.
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| 46. | The integrand in ( 10 ) has sin ( ? ) factor which disappears in the discrete expression ( 11 ).
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| 47. | Functionally integrating over all configurations over counts the integrand repeatedly an infinite number of times giving us the non-sensical result
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| 48. | The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used.
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| 49. | For each small region, the value of the integrand cannot vary much so it may be replaced by a single value.
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| 50. | The two functions " f " and " g " are respectively called the integrand and the integrator.
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