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Cálculo de integrais do tipo∫ ( f ⋅ f ) dx αL βL b L
f a L
f
L
b
L
b
b L
b1 L
b2
a ⋅b ⋅ L
1 ⋅ a ⋅b ⋅ L 2 b⋅L ⋅ (a1 + 2 ⋅ a2 ) 6 a ⋅b ⋅ L ⋅ (1 + α ) 6 1 ⋅ a ⋅b ⋅ L 3 1 ⋅ a ⋅b ⋅ L 4 5 ⋅ a ⋅b ⋅ L 12 b⋅L ⋅ (a2 + 2 ⋅ a3 ) 6
1 ⋅ a ⋅b ⋅ L 2 b⋅L ⋅ (2 ⋅ a1 + a2 ) 6 a ⋅b ⋅ L ⋅ (1 + β ) 6 1 ⋅ a ⋅b ⋅ L 3 1 ⋅ a ⋅b⋅ L 12 1 ⋅ a ⋅b ⋅ L 4 b⋅ L ⋅ (a1 + 2 ⋅ a3 ) 6
a⋅L ⋅ (b1 + b2 ) 2
L ⋅ (2 ⋅ a1 ⋅ b1 + a1 ⋅ b2 + a2 ⋅ b1 + 2 ⋅ a2 ⋅ b2 ) 6 a⋅L ⋅ [(1 + β ) ⋅ b1 + (1 + α ) ⋅ b2 ] 6 1 ⋅ a ⋅ L ⋅ (b1 + b2 ) 3 a⋅L ⋅ (b1 + 3 ⋅ b2 ) 12 a⋅L ⋅ (3 ⋅ b1 + 5 ⋅ b2 ) 12 L ⋅ [a1 ⋅ b1 + a2 ⋅ b2 + 2 ⋅ a3 ⋅ (b1 + b2 )] 6
1 ⋅ a ⋅b ⋅ L 2 b⋅L ⋅ [(1 + β ) ⋅ a1 + (1 + α ) ⋅ a2 ] 6 1 ⋅ a ⋅b ⋅ L 3 a ⋅b ⋅ L ⋅ (1 + α ⋅ β ) 3 a ⋅b ⋅ L ⋅ 1+α +α 2 12
a1 L
a2
b⋅L ⋅ (a1 + a2 ) 2 1 ⋅ a ⋅b ⋅ L 2
*
αL a L
βL
a L
2 ⋅ a ⋅b ⋅ L 3 1 ⋅ a ⋅b ⋅ L 3 2 ⋅ a ⋅b ⋅ L 3 b⋅L ⋅ (a1 + a2 + 4 ⋅ a3 ) 6
tangente
a L
*
(
) )
tangente
a L
L/2 a1 a3 L a2
*
a ⋅b ⋅ L ⋅ 5− β − β 2 12
(
*
∫ ( f ⋅ f ) dx
FEUP
* parábola de segundo grau