sol7 - MAE294B/SIO203B: Methods in Applied Mechanics Winter...

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Unformatted text preview: MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Solution VII 1 The function h ( t ) =- t 4 ( 1- t ) 2 has maxima at the endpoints 0 and 1 of the interval. The contribution from t = 1 is the standard local maximum times 1 / 2. The contribution from t = is different since h 00 ( ) = 0 and the prefactor t vanishes. One can proceed simply by expanding locally and writing I Z t e- xt 4 d t + Z 1- e- x ( 1- t ) 2 d t = 3 4 r x . 2 The function h =- t 3 + 3 t 2- 2 t is a cubic with a maximum at t = 1 + 1 / 3 in the range of integration. Expanding gives the exact result h ( t ) = h ( t )+ 1 2 h 00 ( t )( t- t ) 2 + 1 6 h 000 ( t )( t- t ) 3 = h ( t )- 3 ( t- t ) 2- ( t- t ) 3 . Substitute in: I = Z e x [ h ( t )- 3 ( t- t ) 2 ] 1- x ( t- t ) 3 + x 2 2 ( t- t ) 6 + d t . The usual change of variable combined with extending the range gives I e 2 x / 3 3 1 / 2 3- 1 / 4 x- 1 / 2 1 + 5 16 3 x + O (...
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sol7 - MAE294B/SIO203B: Methods in Applied Mechanics Winter...

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