It seems pretty frustrating that some definite integrals can only be evaluated numerically, so have people tried to find another method of finding the area under a curve that isn't numerical? I'm talking about a method that gives a function that can be evaluated in order to find the area. Is there only one such function for a given curve (the function given by integrating)? Thanks.
2026-04-06 14:48:48.1775486928
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Have people tried to find the area under a curve by means other than integration?
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Yes. There are some curves that form figures that have known geometric properties. Here are some examples:
$$y=\sqrt{1-x^2}$$
$$A=\frac{1}{2}\pi{r}^2=\frac{1}{2}\pi$$
$$y=x$$
From the equation for the area of a right triangle $A=\frac{1}{2}ba$, the area under the curve from $0$ to $x=b$:
$$A=\frac{1}{2}b^2$$
There are many more of such figures (some curves form trapezoids, rectangles, ellipses, etc), which have methods of calculating their area that long predate the fundamental theorem of calculus.
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The one-word answer to your question is "yes". If you chase all the links previous to 1665 in this Wikipedia article, you'll have yourself a good history of the subject: http://en.wikipedia.org/wiki/Timeline_of_calculus_and_mathematical_analysis
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I'm not 100% sure that this argument holds, so please comment on it.
Let's say you have a curve for which there is no exact formula for the anti-derivative. Let's say you just found a magic way to come up with an exact formula that calculates the area anyway (the formula must have as parameters the end points of the independent variable, or you cheat yourself from your own constraint of the method being general).
The fundamental theorem of calculus is a statement about integrals and derivatives; it doesn't care whether there is a nice formula for it or not. If your formula calculates the area, it must be equal to the anti-derivative evaluated at the second point of the domain, minus the anti-derivative at the first point.
I venture - then - that you can use your magic formula for the area to come up with an exact formula for the anti-derivative. At worst, the formula will be an implicit equation, but it will be there.
Therefore, either you were wrong about there not being a non-numerical formula for the anti-derivative, or there is no such method to find a formula for the area when there is no exact formula for the anti-derivative.
I'm not saying that there are no 'non-calculus' ways to come up with a formula for some curves; they exist for (a subset of) the cases where a formula for the anti-derivative exists. I'm saying you can't use those methods to get around the cases where there is no exact formula for the anti-derivative.
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Monte Carlo integration methods can be adapted to be "non-numerical." The idea boils down to randomly sampling a curve bounded by a much simpler one (e.g., a rectangle) at many discrete points. One then counts how many samples were inside the curve and how many were outside. Knowing the area of the simpler bounding shape, the area of (or under) the curve can be estimated.
The classic example of this is throwing darts (but it could be dropping pins or marbles) at a circle inscribed within a square. Often simple scheme is used to estimate $\pi$ (very inefficiently), but it of course also estimates the area of the circle:
Monte Carlo techniques, in their numerical forms, are important for things such as rejection sampling to obtain random numbers from a desired distribution.
It depends on what we mean by "integrating."
There were many quadratures, both ancient and early modern, that did not involve the calculation of antiderivatives in the first-year calculus sense. By the middle of the seventeenth century, all curves of the shape $y=x^\alpha$ had in principle been dealt with, along with a number of others, such as the cycloid. One might mention in this context the Method of Fermat, and the indivisibles of Torricelli and Cavalieri. This was all quite a number of years before the official appearance of the calculus at the hands of Newton and Leibniz.