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Integration & differentiation 
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#1
Dec206, 04:25 PM

P: 5

Under what condition can we change the order of integration and differentiation?
Thanks! 


#2
Dec206, 04:43 PM

P: 1,236

What do you mean by "change"?



#3
Dec206, 05:00 PM

P: 4,006

Changing the order of integration is done to facilitate the actual integration, ie the integrand and the equation of the boundaries. marlon 


#4
Dec306, 06:07 AM

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PF Gold
P: 12,016

Integration & differentiation
Eeh, NO, marlon!
To take a trivial example, have a continuous, but nondifferentiable integrand. An antiderivative of this function is certainly differentiable, and yields back the integrand, by FOTC. However, since your integrand is nondifferentiable, you cannot differentiate it first, and then compute that nonexistent function's antiderivative. The upshot of this is that you may change the order of differentiation/integration as long as your integrand is sufficiently nice. 


#6
Dec306, 01:15 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,547

Libnitz's formula: If [itex]\phi (x,t)[/itex] is continuous in t and differentiable in x, then
[tex]\frac{d }{dx}\int_{\alpha (x)}^{\beta(x)} \phi(x,t)dt= \frac{d\alpha}{dx}\phi(x,\alpha(x)) \frac{d\beta}{dt}\phi(x,\beta(x))+ \int_{\alpha (x)}^{\beta(x)} \frac{\partial \phi}{\partial x} dt[/itex] In particular, if the limits of integration are constant, then [tex]\frac{d }{dx}\int_a^b \phi(x,t)dt= \int_a^b\frac{\partial \phi}{\partial x}dt[/tex] 


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