An ordinary integral is another name for the plain old single integral, consisting of:
- A function to be integrated (an integrand) over the real (numbers) line,
- One variable to integrate over (e.g. x or y),
- A region of space to integrate over (a domain of integration).
Another way to define an ordinary integral: It’s the only one where you can use the fundamental theorem of calculus to find a solution. With other types, you can’t directly use the theorem; You have to manipulate the integral in some way first.
Comparison of Ordinary Integral to Other Integrals
Although it’s usually called just an “integral” in introductory calculus, it’s referred to as “ordinary” in higher calculus texts to set it apart from other integrals, like:
- Line integrals (generalized ordinary integrals that work along curves),
- Product integrals (which have multiplication instead of summation),
- Double integrals (a stack of two single integrals). With an ordinary (single) integral, you can use the fundamental theorem of calculus to integrate over a region. A 2-dimensional double integral has to be broken apart into two 1-dimensional ordinary integrals before it can be evaluated (Goetz, 2020).
Why are they called “Ordinary”?
Ordinary integrals are called ordinary because they are equal to ordinary differential equations (Hall, 2012). For example, the following ordinary differential equation, with initial condition X(0) = x0
is equivalent to finding a solution to the following ordinary integral (Papaspiliopoulos, 2007):
Goetz, P. (2020). LM15-2-2. Retrieved September 1, 2020 from: https://web.ma.utexas.edu/users/m408s/m408d/current/LM15-2-2-pgoetz.html
Hall, W. (2012). The Boundary Element Method. Springer Netherlands.
Papaspiliopoulos, A. (2007). An introduction to modelling and likelihood inference with stochastic differential equations. Retrieved September 1, 2020 from: http://126.96.36.199/~omiros/course_notes.pdf
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