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You should use numerical_integral:

sage: f(x) = 1/(3.33*sqrt(2*pi))*exp(-1/2*((x-9.486)/3.33)^2)                   
sage: numerical_integral(f, (6,23))                                             
(0.8523901699416254, 9.463431926333425e-15)

The second number in the output is an estimate of the numerical error (type numerical_intergal? for more details, in particular for options to control the numerical error).