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  • Writer's pictureNicholas Masagao

Improper Integrals

Improper integrals are integrals that do not converge in the usual sense. These types of integrals are important in mathematics, engineering, and physics, as they allow us to calculate values for functions that would otherwise be difficult or impossible to obtain. In this blog post, we will explore what improper integrals are, how to evaluate them, and some examples of their applications.


Lecture 6.6
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What are Improper Integrals?


An improper integral is an integral that cannot be evaluated using standard methods.


This can occur for a few reasons, including:

  • The integrand is undefined at one or more points in the interval of integration.

  • The interval of integration is infinite.

  • The integrand is unbounded on the interval of integration.

When we encounter these situations, we must take a different approach to evaluating the integral. We can do this by either breaking up the integral into smaller pieces or by taking the limit as the interval of integration approaches infinity.


Evaluating Improper Integrals


To evaluate an improper integral, we can use one of two methods: breaking up the integral into smaller pieces or taking the limit as the interval of integration approaches infinity.


Breaking Up the Integral


If the integrand is undefined at one or more points in the interval of integration, we can break up the integral into smaller pieces. For example, consider the integral:

∫(1 / (x - 1)) dx from 0 to 2

The integrand is undefined at x = 1. Therefore, we can break up the integral into two smaller integrals:

∫(1 / (x - 1)) dx from 0 to 1-ε and ∫(1 / (x - 1)) dx from 1+ε to 2

where ε is a small positive number.


Taking the Limit


If the interval of integration is infinite or the integrand is unbounded on the interval of integration, we must take the limit as the interval of integration approaches infinity. For example, consider the integral:


∫(1 / x^2) dx from 1 to infinity


The integrand is unbounded at x = 0, and the interval of integration is infinite. Therefore, we must take the limit as the upper limit of integration approaches infinity:

lim as b -> infinity ∫(1 / x^2) dx from 1 to b

= lim as b -> infinity (-1 / x) from 1 to b

= 1


Applications of Improper Integrals


Improper integrals have many applications in mathematics, engineering, and physics. Some common examples include:

  • Calculating areas under curves that have infinite length, such as the curve y = 1/x^2.

  • Calculating the volume of objects with infinite length or radius, such as a cylinder with infinite length.

  • Calculating the probability of events that occur in continuous time or space, such as the probability of a particle being in a certain position at a certain time.


Conclusion


Improper integrals are an important tool in mathematics, engineering, and physics. They allow us to calculate values for functions that would otherwise be difficult or impossible to obtain. By breaking up the integral into smaller pieces or taking the limit as the interval of integration approaches infinity, we can evaluate improper integrals and apply them to a variety of real-world problems.

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