Order of Integration and Differentiation: Are There Any Exceptions?

[helenwang413]

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

Thanks!

 

[courtrigrad]

What do you mean by "change"?

 

[marlon]

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

Thanks!

The order of integration can be changed always. If i remember correctly, that's one of the key requirements for an integral to exist.

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

marlon

 

[arildno]

Eeh, NO, marlon!

To take a trivial example, have a continuous, but non-differentiable integrand.
An anti-derivative of this function is certainly differentiable, and yields back the integrand, by FOTC.

However, since your integrand is non-differentiable, you cannot differentiate it first, and then compute that non-existent function's anti-derivative.

The upshot of this is that you may change the order of differentiation/integration as long as your integrand is sufficiently nice.

 

[marlon]

Eeh, NO, marlon!

To take a trivial example, have a continuous, but non-differentiable integrand.
An anti-derivative of this function is certainly differentiable, and yields back the integrand, by FOTC.

However, since your integrand is non-differentiable, you cannot differentiate it first, and then compute that non-existent function's anti-derivative.

The upshot of this is that you may change the order of differentiation/integration as long as your integrand is sufficiently nice.

Oppps, thanks for the correction.

marlon

 

[HallsofIvy]

Libnitz's formula: If $\phi (x,t)$ 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
$$\frac{d }{dx}\int_a^b \phi(x,t)dt= \int_a^b\frac{\partial \phi}{\partial x}dt$$