In math, an integral assigns values to functions that help complete other functions. For example, to calculate the amount of water a boat displaces, an integral would give value to the area of the boat. This term is used in the branch of math called calculus and comes from the word integration. This is because it integrates data into an equation that would otherwise be impossible to solve.
Definite integrals are a value that falls between certain criteria. This type of integral falls between the upper and lower limits of a variable that is independent of a statement. In other words, it’s a variable that could have a value between two limits. An example of this would be a maximum size or minimum size of the hull of the boat described above.
An improper integral has a much wider limit to its value. It could either be infinite or have the value of integrand, which is an integral that approaches infinite. This concept is a little more challenging to deal with than most integrals since the value could fall between much wider criteria.
Calculating Improper Integrals
The trickiest part of calculating an improper integral is that it isn’t really calculated, but rather analyzed in an equation independent from the function. They are represented symbolically as a limit of a form.
Because there is no definite value to the integral, this is technically abuse of notation. As bad as this may sound, it’s simply a way of using notation to explain something that may not have a formal type of notation, such as a definite integral that may or may not fall somewhere between a fixed value and notation.
The limitations of integrals are what sets them apart. If an integral has two limitations that are clearly defined it’s a definite integral. If one or both limitations may approach infinity, it’s an improper integral.
The notation of these problems will show that one is solvable to find an asymptote with a definite value, while the other cannot provide a value that does not approach infinite. In other words, the first problem can be solved, while the other can only be analyzed.
Convergence and Divergence
Of the two examples above, one is unsolvable, but that doesn’t mean there isn’t data to be found in the analysis. If an improper integral is found to have a finite value it converges. If it is found that both limits approach infinity, it diverges. This means that a starting point can be found in the asymptote if the function is to be graphed. It may not be much, but this could provide a solution to the function the integral provides value for.
What’s most confusing about this type of analysis is that there are still values that can be given by analyzing the infinite limits. It’s all about adding perspective. Because the two values are infinite, there is no middle. Instead, the point of x axis could be considered the middle. Hopefully this will be enough to solve the question of divergence if the integral is to be solved for.