1. Proper Integrals
A proper integral is defined when the limits of integration are finite and the function remains continuous throughout the interval.
An integral is called improper if at least one of the following conditions is satisfied:
One or both integration limits are infinite.
The integrand is undefined or becomes unbounded at some point within the interval.
3. Definition Using Limits
Improper integrals are evaluated by expressing them as limits.
This approach allows us to determine whether the integral converges or diverges.
4. Physical Applications
Improper integrals frequently appear in physics.
In electrostatics, they are used to compute electric fields generated by charge distributions extending to infinity.
In thermodynamics, they help calculate the total energy of systems occupying unbounded regions of space.
5. Convergence and Divergence of Improper Integrals
An improper integral with infinite limits does not always diverge.
Its behavior depends on how the integrand behaves as � becomes large.
If the function decreases sufficiently fast, the integral converges.
If the function decreases too slowly or remains large, the integral diverges.