IMPROPER INTEGRAL Meaning and
Definition
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An improper integral is a mathematical concept used in calculus to evaluate the integral of a function over an unbounded interval, or a bounded interval where the integrand itself becomes infinite or undefined at one or both endpoints. It is termed "improper" because it deviates from the typical notion of a definite integral, which is defined over a bounded interval and integrates a continuous function.
In more specific terms, an improper integral occurs when either the lower or upper limit of integration is infinity or negative infinity, or when the integrand diverges or becomes undefined at one or both endpoints. To compute an improper integral, the function is first integrated over a definite interval, and then the limits are taken to infinity or negative infinity, or the undefined endpoint is approached.
The evaluation of an improper integral involves handling divergent or undefined integrands delicately using precise mathematical methods, such as employing limits or techniques like integration by parts or substitution. By integrating over unbounded intervals or intervals where the integrand is not well-behaved, improper integrals provide a powerful tool for calculating areas, volumes, and other quantitative properties that cannot be determined by ordinary integrals.
Common Misspellings for IMPROPER INTEGRAL
- umproper integral
- jmproper integral
- kmproper integral
- omproper integral
- 9mproper integral
- 8mproper integral
- inproper integral
- ikproper integral
- ijproper integral
- imoroper integral
- imlroper integral
- im0roper integral
- impeoper integral
- impdoper integral
- impfoper integral
- imptoper integral
- imp5oper integral
- imp4oper integral
- impriper integral
- imprkper integral
Etymology of IMPROPER INTEGRAL
The word "improper" in the term "improper integral" refers to the fact that these integrals do not meet the usual conditions for convergence and therefore have a less strict behavior compared to regular integrals.
The term "improper" comes from the Latin word "improprius", which translates to "not proper" or "not appropriate". In mathematics, this adjective is used to describe things that do not fit into the standard framework or rules. In the case of integrals, an improper integral occurs when either the limits of integration are infinite or the function being integrated has a discontinuity or infinite value within the interval of integration. These features make these integrals unconventional or "not proper" in the traditional sense.
