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helenwang413
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Under what condition can we change the order of integration and differentiation?
Thanks!
Thanks!
helenwang413 said:Under what condition can we change the order of integration and differentiation?
Thanks!
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.
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.
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.
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.
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.
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.