Order of Integration and Differentiation: Are There Any Exceptions?

In summary: Hence, if you have a continuous function and you know its limits of integration, you can integrate it at those limits. marlonOppps, thanks for the correction.If \phi (x,t) is continuous in t and differentiable in x, then \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
  • #1
helenwang413
5
0
Under what condition can we change the order of integration and differentiation?

Thanks!
 
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  • #2
What do you mean by "change"?
 
  • #3
helenwang413 said:
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
 
  • #4
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.
 
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  • #5
arildno said:
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
 
  • #6
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]
 

Related to Order of Integration and Differentiation: Are There Any Exceptions?

What is the difference between integration and differentiation?

Integration and differentiation are two fundamental concepts in calculus. Integration is the process of finding the area under a curve while differentiation is the process of finding the slope of a curve. In other words, integration is the reverse process of differentiation.

What are the applications of integration and differentiation in real life?

Integration and differentiation have various applications in fields such as physics, engineering, economics, and finance. For example, integration can be used to calculate the volume of irregularly shaped objects, while differentiation can be used to determine the velocity and acceleration of moving objects.

What are the different methods of integration and differentiation?

There are several techniques for integration, including the fundamental theorem of calculus, substitution, integration by parts, and partial fractions. Similarly, differentiation has various methods such as the power rule, product rule, quotient rule, and chain rule.

How is integration and differentiation related to each other?

Integration and differentiation are inverse operations of each other. This means that if we integrate a function and then differentiate the resulting function, we will get back the original function. This relationship is known as the fundamental theorem of calculus.

What are the practical uses of integration and differentiation in scientific research?

Integration and differentiation are essential tools for solving complex mathematical problems in various scientific fields. Scientists use these concepts to model and analyze real-world phenomena, make predictions, and formulate scientific theories. They are also crucial in the development of new technologies and advancements in various fields.

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